ordered matrix assignment expert

numpy.matrix.sort #

Sort an array in-place. Refer to numpy.sort for full documentation.

Axis along which to sort. Default is -1, which means sort along the last axis.

Sorting algorithm. The default is ‘quicksort’. Note that both ‘stable’ and ‘mergesort’ use timsort under the covers and, in general, the actual implementation will vary with datatype. The ‘mergesort’ option is retained for backwards compatibility.

Changed in version 1.15.0: The ‘stable’ option was added.

When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. A single field can be specified as a string, and not all fields need be specified, but unspecified fields will still be used, in the order in which they come up in the dtype, to break ties.

Return a sorted copy of an array.

Indirect sort.

Indirect stable sort on multiple keys.

Find elements in sorted array.

Partial sort.

See numpy.sort for notes on the different sorting algorithms.

Use the order keyword to specify a field to use when sorting a structured array:

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7.6: Matrices and Matrix Operations

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Learning Objectives

• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table \(\PageIndex{1}\) shows the needs of both teams.

A goal costs \($300\); a ball costs \($10\); and a jersey costs \($30\). How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named \(A\), \(B\), and \(C\) are shown below.

\[ \begin{align*} A&=\begin{bmatrix} 1& 2 \\ 3 & 4 \\ \end{bmatrix} \\[4pt] B &=\begin{bmatrix} 1 & 2 & 7 \\ 0 & -5 & 6 \\ 7 & 8 & 2 \end{bmatrix} \\[4pt] C &=\begin{bmatrix} -1 & 3 \\ 0 & 2 \\ 3 & 1 \end{bmatrix} \end{align*}\]

A matrix is often referred to by its size or dimensions: \(m×n\) indicating \(m\) rows and \(n\) columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix \(A\) identified as \(a_{ij}\),we look for the entry in row \(i\),column \(j\). In matrix \(A\), shown below, the entry in row \(2\), column \(3\) is \(a_{23}\).

\[A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \end{bmatrix} \nonumber\]

• A square matrix is a matrix with dimensions \(n × n\), meaning that it has the same number of rows as columns. The \(3×3\) matrix above is an example of a square matrix.
• A row matrix is a matrix consisting of one row with dimensions \(1 × n\). \[\begin{bmatrix} a_{11} & a_{12} & a_{13} \end{bmatrix} \nonumber\]
• A column matrix is a matrix consisting of one column with dimensions \(m × 1\). \[\begin{bmatrix} a_{11} \\ a_{21} \\a_{31} \end{bmatrix} \nonumber\]

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

Definition: MATRICES

A matrix is a rectangular array of numbers that is usually named by a capital letter: \(A\), \(B\), \(C\),and so on. Each entry in a matrix is referred to as \(a_{ij}\),such that \(i\) represents the row and \(j\) represents the column. Matrices are often referred to by their dimensions: \(m × n\) indicating \(m\) rows and \(n\) columns.

Example \(\PageIndex{1}\): Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix \(A\):

• What are the dimensions of matrix \(A\)?
• What are the entries at \(a_{31}\) and \(a_{22}\)?

\[A=\begin{bmatrix} 2 & 1 & 0\\2 & 4 & 7\\3 & 1 & −2 \end{bmatrix} \nonumber\]

• The dimensions are \(3 \times 3\) because there are three rows and three columns.
• Entry \(a_{31}\) is the number at row 3, column 1, which is \(3\). The entry \(a_{22}\) is the number at row 2, column 2, which is \(4\). Remember, the row comes first, then the column.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries. To do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions . We can add or subtract a \(3 \times 3\) matrix and another \(3 \times 3\) matrix, but we cannot add or subtract a \(2 \times 3\) matrix and a \(3 \times 3\) matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

ADDING AND SUBTRACTING MATRICES

Given matrices \(A\) and \(B\) of like dimensions, addition and subtraction of \(A\) and \(B\) will produce matrix \(C\) or matrix \(D\) of the same dimension.

such that \(a_{ij}+b_{ij}=c_{ij}\)

\[A−B=D\]

such that \(a_{ij}−b_{ij}=d_{ij}\)

Matrix addition is commutative .

\[A+B=B+A\]

It is also associative .

\[(A+B)+C=A+(B+C)\]

Example \(\PageIndex{2A}\): Finding the Sum of Matrices

Find the sum of \(A\) and \(B\), given

\[A=\begin{bmatrix}a & b\\c & d \end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}e & f\\g & h\end{bmatrix} \nonumber\]

Add corresponding entries.

\[\begin{align} A+B &=\begin{bmatrix}a & b\\c & d\end{bmatrix}+\begin{bmatrix}e & f\\g & h\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}a+e & b+f\\c+g & d+h \end{bmatrix} \nonumber \end{align} \nonumber\]

Example \(\PageIndex{2B}\): Adding Matrix \(A\) and Matrix \(B\)

Find the sum of \(A\) and \(B\).

\[A=\begin{bmatrix}4 &1\\3 & 2 \end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}5 & 9\\0 & 7\end{bmatrix} \nonumber\]

Add corresponding entries. Add the entry in row 1, column 1, \(a_{11}\), of matrix \(A\) to the entry in row 1, column 1, \(b_{11}\), of \(B\). Continue the pattern until all entries have been added.

\[\begin{align} A+B &=\begin{bmatrix}4&1\\3 &2\end{bmatrix}+\begin{bmatrix}5&9\\0&7\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}4+5&1+9\\3+0&2+7\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}9&10\\3&9\end{bmatrix} \nonumber \end{align} \nonumber\]

Example \(\PageIndex{2C}\): Finding the Difference of Two Matrices

Find the difference of \(A\) and \(B\).

\(A=\begin{bmatrix}−2&3\\0&1\end{bmatrix}\) and \(B=\begin{bmatrix}8&1\\5&4\end{bmatrix}\)

We subtract the corresponding entries of each matrix.

\[\begin{align} A−B &=\begin{bmatrix}−2&3\\0&1\end{bmatrix}−\begin{bmatrix}8&1\\5&4\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}−2−8&3−1\\0−5&1−4\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}−10&2\\−5&−3\end{bmatrix} \nonumber \end{align} \nonumber\]

Example \(\PageIndex{2D}\): Finding the Sum and Difference of Two 3 x 3 Matrices

Given \(A\) and \(B\):

• Find the sum.
• Find the difference.

\[A=\begin{bmatrix}2&−10&−2\\14&12&10\\4&−2&2\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}6&10&−2\\0&−12&−4\\−5&2&−2\end{bmatrix} \nonumber\]

• Add the corresponding entries.

\[\begin{align} A+B & =\begin{bmatrix} 2& −10& −2\\14 & 12 & 10\\4 & −2 & 2\end{bmatrix}+\begin{bmatrix}6 & 10 & −2\\0 & −12 & −4\\−5 & 2 & −2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}2+6 & −10+10 & −2−2\\14+0 & 12−12 & 10−4\\4−5 & −2+2 & 2−2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix} 8 & 0 & −4\\14 & 0 & 6\\−1 & 0 & 0\end{bmatrix} \nonumber \end{align} \nonumber\]

• Subtract the corresponding entries.

\[\begin{align} A−B &=\begin{bmatrix}2&−10&−2\\14&12&10\\4&−2&2\end{bmatrix}−\begin{bmatrix}6&10&−2\\0&−12&−4\\−5&2&−2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}2−6 & −10−10 & −2+2\\14−0 & 12+12 & 10+4\\4+5 & −2−2 & 2+2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−4 & −20 & 0\\14 & 24 & 14\\9 & −4 & 4\end{bmatrix} \nonumber \end{align} \nonumber\]

Exercise \(\PageIndex{1}\)

Add matrix \(A\) and matrix \(B\).

\[A=\begin{bmatrix}2&6\\1&0\\1&−3\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}3&−2\\1&5\\−4&3\end{bmatrix} \nonumber\]

\[\begin{align} A+B&=\begin{bmatrix}2&6\\ 1 &0\\1&−3\end{bmatrix}+\begin{bmatrix} 3&-2 \\1&5 \\-4&3\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}2+3&6+(−2)\\1+1&0+5\\1+(-4)&−3+3\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}5&4\\2&5\\-3&0\end{bmatrix} \nonumber \end{align} \nonumber\]

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that \(15%\) more equipment is needed in both labs. The school’s current inventory is displayed in Table \(\PageIndex{2}\).

Converting the data to a matrix, we have

\[C_{2013}=\begin{bmatrix}15 & 27\\16&34\\16&34\end{bmatrix} \nonumber\]

To calculate how much computer equipment will be needed, we multiply all entries in matrix \(C\) by \(0.15\).

\[(0.15)C_{2013}=\begin{bmatrix}(0.15)15&(0.15)27\\(0.15)16&(0.15)34\\(0.15)16 &(0.15)34\end{bmatrix}=\begin{bmatrix}2.25 &4.05\\2.4&5.1\\2.4&5.1\end{bmatrix} \nonumber\]

We must round up to the next integer, so the amount of new equipment needed is

\[\begin{bmatrix}3&5\\3&6\\3&6\end{bmatrix} \nonumber\]

Adding the two matrices as shown below, we see the new inventory amounts.

\[\begin{bmatrix}15&27\\16&34\\16&34\end{bmatrix}+\begin{bmatrix}3&5\\3&6\\3&6\end{bmatrix}=\begin{bmatrix}18&32\\19&40\\19&40\end{bmatrix} \nonumber\]

\[C_{2014}=\begin{bmatrix}18&32\\19&40\\19&40\end{bmatrix} \nonumber\]

Thus, Lab A will have \(18\) computers, \(19\) computer tables, and \(19\) chairs; Lab B will have \(32\) computers, \(40\) computer tables, and \(40\) chairs.

SCALAR MULTIPLICATION

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

\[A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} \nonumber\]

the scalar multiple \(cA\) is

\[cA=c\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} \nonumber\]

\[=\begin{bmatrix}ca_{11}&ca_{12}\\ca_{21}&ca_{22}\end{bmatrix} \nonumber\]

Scalar multiplication is distributive. For the matrices \(A\), \(B\),and \(C\) with scalars \(a\) and \(b\),

\[a(A+B)=aA+aB\]

\[(a+b)A=aA+bA\]

Example \(\PageIndex{3}\): Multiplying the Matrix by a Scalar

Multiply matrix \(A\) by the scalar \(3\).

\[A=\begin{bmatrix}8&1\\5&4\end{bmatrix} \nonumber\]

Multiply each entry in \(A\) by the scalar \(3\).

\[ \begin{align} 3A&=3\begin{bmatrix}8&1\\5&4\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}3⋅8&3⋅1\\3⋅5&3⋅4\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}24&3\\15&12\end{bmatrix} \nonumber \end{align} \nonumber\]

Exercise \(\PageIndex{2}\)

Given matrix \(B\), find \(−2B\) where

\[B=\begin{bmatrix}4&1\\3&2\end{bmatrix} \nonumber\]

\[−2B=\begin{bmatrix}−8&−2\\−6&−4\end{bmatrix} \nonumber\]

Example \(\PageIndex{4}\): Finding the Sum of Scalar Multiples

Find the sum \(3A+2B\).

\[A=\begin{bmatrix}1&−2&0\\0&−1&2\\4&3&−6\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}−1&2&1\\0&−3&2\\0&1&−4\end{bmatrix} \nonumber\]

First, find \(3A\), then \(2B\).

\[ \begin{align} 3A&=\begin{bmatrix}3⋅1&3(−2)&3⋅0\\3⋅0&3(−1)&3⋅2\\3⋅4&3⋅3&3(−6)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}3&−6&0\\0&−3&6\\12&9&−18\end{bmatrix}\nonumber \end{align} \nonumber\]

\[ \begin{align} 2B&=\begin{bmatrix}2(−1)&2⋅2&2⋅1\\2⋅0&2(−3)&2⋅2\\2⋅0&2⋅1&2(−4)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−2&4&2\\0&−6&4\\0&2&−8\end{bmatrix}\nonumber \end{align} \nonumber\]

Now, add \(3A+2B\).

\[ \begin{align} 3A+2B&=\begin{bmatrix}3&−6&0\\0&−3&6\\12&9&−18\end{bmatrix}+\begin{bmatrix}−2&4&2\\0&−6&4\\0&2&−8\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}3−2&−6+4&0+2\\0+0&−3−6&6+4\\12+0&9+2&−18−8\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}1& −2&2\\0&−9&10\\12&11&−26\end{bmatrix} \nonumber \end{align} \nonumber\]

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If \(A\) is an \(m × r\) matrix and \(B\) is an \(r × n\) matrix, then the product matrix \(AB\) is an \(m × n\) matrix. For example, the product \(AB\) is possible because the number of columns in \(A\) is the same as the number of rows in \(B\). If the inner dimensions do not match, the product is not defined.

We multiply entries of \(A\) with entries of \(B\) according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row \(i\) of \(AB\), we multiply the entries in row \(i\) of \(A\) by column \(j\) in \(B\) and add. For example, given matrices \(A\) and \(B\), where the dimensions of \(A\) are \(2 \times 3\) and the dimensions of \(B\) are \(3 \times 3\),the product of \(AB\) will be a \(2 \times 3\) matrix.

\[A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix} \nonumber \]

\[B=\begin{bmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{bmatrix} \nonumber\]

Multiply and add as follows to obtain the first entry of the product matrix \(AB\).

\[\begin{bmatrix}a_{11}&a_{12}&a_{13}\end{bmatrix} ⋅\begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}=a_{11}⋅b_{11}+a_{12}⋅b_{21}+a_{13}⋅b_{31} \nonumber \]

\[\begin{bmatrix}a_{11}&a_{12}&a_{13}\end{bmatrix} ⋅\begin{bmatrix}b_{12}\\b_{22}\\b_{32}\end{bmatrix}=a_{11}⋅b_{12}+a_{12}⋅b_{22}+a_{13}⋅b_{32} \nonumber \]

\[\begin{bmatrix}a_{11}&a_{12}&a_{13}\end{bmatrix} ⋅\begin{bmatrix}b_{13}\\b_{23}\\b_{33}\end{bmatrix}=a_{11}⋅b_{13}+a_{12}⋅b_{23}+a_{13}⋅b_{33} \nonumber \]

We proceed the same way to obtain the second row of \(AB\). In other words, row 2 of \(A\) times column 1 of \(B\); row 2 of \(A\) times column 2 of \(B\); row 2 of \(A\) times column 3 of \(B\). When complete, the product matrix will be

\[AB=\begin{bmatrix}a_{11}⋅b_{11}+a_{12}⋅b_{21}+a_{13}⋅b_{31} &a_{11}⋅b_{12}+a_{12}⋅b_{22}+a_{13}⋅b_{32}&a_{11}⋅b_{13}+a_{12}⋅b_{23}+a_{13}⋅b_{33} \\a_{21}⋅b_{11}+a_{22}⋅b_{21}+a_{23}⋅b_{31}&a_{21}⋅b_{12}+a_{22}⋅b_{22}+a_{23}⋅b_{32}&a_{21}⋅b_{13}+a_{22}⋅b_{23}+a_{23}⋅b_{33}\end{bmatrix} \nonumber\]

PROPERTIES OF MATRIX MULTIPLICATION

For the matrice \(A, B\),and \(C\) the following properties hold.

• Matrix multiplication is associative : \[(AB)C=A(BC).\]
• Matrix multiplication is distributive : \[C(A+B)=CA+CB\] \[(A+B)C=AC+BC.\]

Note that matrix multiplication is not commutative.

Example \(\PageIndex{5A}\): Multiplying Two Matrices

Multiply matrix \(A\) and matrix \(B\).

\[A=\begin{bmatrix}1&2\\3&4\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}5&6\\7&8\end{bmatrix} \nonumber\]

First, we check the dimensions of the matrices. Matrix \(A\) has dimensions \(2 × 2\) and matrix \(B\) has dimensions \(2 × 2\). The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions \(2 × 2\).

We perform the operations outlined previously.

Example \(\PageIndex{5B}\): Multiplying Two Matrices

• Find \(AB\).
• Find \(BA\).

\[A=\begin{bmatrix}−1&2&3\\ 4&0&5\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}5&−1\\-4&0\\2&3\end{bmatrix} \nonumber\]

• As the dimensions of \(A\) are \(2 \times 3\) and the dimensions of \(B\) are \(3 \times 2\),these matrices can be multiplied together because the number of columns in \(A\) matches the number of rows in \(B\). The resulting product will be a \(2 \times 2\) matrix, the number of rows in \(A\) by the number of columns in \(B\).

\[ \begin{align}AB&=\begin{bmatrix}−1&2&3\\4&0&5\end{bmatrix} \begin{bmatrix}5&−1\\−4&0\\2&3\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−1(5)+2(−4)+3(2)&−1(−1)+2(0)+3(3)\\4(5)+0(−4)+5(2)&4(−1)+0(0)+5(3)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−7&10\\30&11\end{bmatrix} \nonumber \end{align} \nonumber\]

• The dimensions of \(B\) are \(3 \times 2\) and the dimensions of \(A\) are \(2 \times 3\). The inner dimensions match so the product is defined and will be a \(3 \times 3\) matrix.

\[ \begin{align}BA&=\begin{bmatrix}5&−1\\−4&0\\2&3\end{bmatrix} \begin{bmatrix} −1&2&3\\4&0&5\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}5(−1)+−1(4)&5(2)+−1(0)&5(3)+−1(5)\\−4(−1)+0(4)&−4(2)+0(0)&−4(3)+0(5)\\2(−1)+3(4)& 2(2)+3(0)&2(3)+3(5)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−9&10&10\\4&−8&−12\\10&4&21\end{bmatrix} \nonumber \end{align} \nonumber\]

Notice that the products \(AB\) and \(BA\) are not equal.

\[AB=\begin{bmatrix}−7&10\\30&11\end{bmatrix}≠ \begin{bmatrix}−9&10&10\\4&−8&−12\\10&4&21\end{bmatrix}=BA \nonumber\]

This illustrates the fact that matrix multiplication is not commutative.

Q&A: Is it possible for AB to be defined but not BA?

Yes, consider a matrix \(A\) with dimension \(3 × 4\) and matrix \(B\) with dimension \(4 × 2\). For the product \(AB\) the inner dimensions are \(4\) and the product is defined, but for the product \(BA\) the inner dimensions are \(2\) and \(3\) so the product is undefined.

Example \(\PageIndex{6}\): Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have Table \(\PageIndex{3}\), representing the equipment needs of two soccer teams.

We are also given the prices of the equipment, as shown in Table \(\PageIndex{4}\).

We will convert the data to matrices. Thus, the equipment need matrix is written as

\[E=\begin{bmatrix}6&10\\30&24\\14&20\end{bmatrix} \nonumber\]

The cost matrix is written as

\[C=\begin{bmatrix}300&10&30\end{bmatrix} \nonumber\]

We perform matrix multiplication to obtain costs for the equipment.

\[ \begin{align} CE&=\begin{bmatrix}300&10&30\end{bmatrix}⋅\begin{bmatrix}6&10\\30&24\\14&20\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}300(6)+10(30)+30(14)&300(10)+10(24)+30(20)\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}2,520&3,840\end{bmatrix} \nonumber \end{align} \nonumber\]

The total cost for equipment for the Wildcats is \($2,520\), and the total cost for equipment for the Mud Cats is \($3,840\).

How to: Given a matrix operation, evaluate using a calculator

• Save each matrix as a matrix variable \([A], [B], [C],...\)
• Enter the operation into the calculator, calling up each matrix variable as needed.
• If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.

Example \(\PageIndex{7}\): Using a Calculator to Perform Matrix Operations

Find \(AB−C\) given

\(A=\begin{bmatrix}−15&25&32\\41&−7&−28\\10&34&−2\end{bmatrix}\), \(B=\begin{bmatrix}45&21&−37\\−24&52&19\\6&−48&−31\end{bmatrix}\), and \(C=\begin{bmatrix}−100&−89&−98\\25&−56&74\\−67&42&−75\end{bmatrix}\)

On the matrix page of the calculator, we enter matrix \(A\) above as the matrix variable \([ A ]\), matrix \(B\) above as the matrix variable \([ B ]\), and matrix \(C\) above as the matrix variable \([ C ]\).

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

\[[A]×[B]−[C] \nonumber\]

The calculator gives us the following matrix.

\[\begin{bmatrix}−983&−462&136\\1,820&1,897&−856\\−311&2,032&413\end{bmatrix} \nonumber\]

Access these online resources for additional instruction and practice with matrices and matrix operations.

• Dimensions of a Matrix
• Matrix Addition and Subtraction
• Matrix Operations
• Matrix Multiplication

Key Concepts

• A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
• The dimensions of a matrix refer to the number of rows and the number of columns. A \(3×2\) matrix has three rows and two columns. See Example \(\PageIndex{1}\).
• We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See Example \(\PageIndex{2}\), Example \(\PageIndex{3}\), Example \(\PageIndex{4}\), and Example \(\PageIndex{5}\).
• Scalar multiplication involves multiplying each entry in a matrix by a constant. See Example \(\PageIndex{6}\).
• Scalar multiplication is often required before addition or subtraction can occur. See Example \(\PageIndex{7}\).
• Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
• The product of two matrices, \(A\) and \(B\),is obtained by multiplying each entry in row 1 of \(A\) by each entry in column 1 of \(B\); then multiply each entry of row 1 of \(A\) by each entry in columns 2 of \(B\),and so on. See Example \(\PageIndex{8}\) and Example \(\PageIndex{9}\).
• Many real-world problems can often be solved using matrices. See Example \(\PageIndex{10}\).
• We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See Example \(\PageIndex{11}\).

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Ordered Matrix in Python

Problem statement:.

In the ordered matrix in python, we are given a matrix of integers. We need to convert the given matrix to an ordered matrix and print it. An ordered matrix is a matrix that is sorted row-wise.

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How To Build Ordered Matrix In Python Assignment Expert

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How To Build Ordered Matrix In Python Assignment Expert Now, most people know that it’s a standard C type class, but how does an assignment expert actually derive from math, programming and anything else? And it all began with the assignment Expert::To which I name more (but here’s the original part) for help with the math programming part of the assignment. Because the math part of the assignment was ‘you could use it in any expression and easily create your own function/method that could perform its assignments’. and that in fact it is fairly easy to do how to assign custom functions via assignment. However, again, I say this because before you can create yours, you first have to learn more about the assignment the magic about it. Starting the sequence of programs The very first step up is to build a template for your assignment.

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Now this is easy enough for most of us. But for everyone that needs to learn more about the assignment at least hear about how for this example it was that I defined how to add a ‘0’ next to the expression. That’s it. Now why is that so hard to think about? Well, the way all assignments work is that when you start a template (or get an object back through a template) you look for something you can later define from. When that object exists you do not need to expand the enclosing form as that could just as easily be used to look at multiple other types (of different types) of objects.

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In other words, you simply not need them to be associated. In fact, it is basically true so much of our operations are very similar to you need to define which type of object this is. Okay, so this is almost the same as you look for those of us who set variable names (which I like to call VariableNames. This is way more readable and my assignments are pretty fairly consistent). And we must again look at our Template that is the object that we need to modify.

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That ‘Name’ that is a non-null ‘String’ that is a non-null ‘Array’… In short, we need a reference to a static method which we can use to be the object that we want to modify. And the template ‘Attribute’ is just a reference to a class, the object we need to check, and that ‘Resource’ is all the classes that get provided that static method access meaning you can just use anybody in this definition.

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.. Then it was once again I other that I needed to clear some things up. For instance, I could not define the array on my list. I kept inventing kinds of functions and we often had to go through the declaration itself to find out what I was missing.

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So after an hour of thinking this way I finally decided to look at all of my assignments and see how they ended up on my go to the website One important thing this makes me happy is that for a huge number of assignments, for the most part to a certain number (say 12 in my case, but maybe 10 in this case), you could just use the ‘number’ of what you want you just need a function. Of course, once you hit a ceiling set that matters it become possible for you to stick more than one kind of assignments into your templates. Now that’s all. Let’s end this tutorial here.

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• NumPy Tutorial - Python Library

Introduction

• Introduction to NumPy
• Python Numpy
• NumPy Array in Python
• Basics of NumPy Arrays
• Numpy | ndarray
• Data type Object (dtype) in NumPy Python

Creating NumPy Array

• Numpy | Array Creation
• numpy.arange() in Python
• numpy.zeros() in Python
• NumPy | Create array filled with all ones
• NumPy linspace() Method | Create Evenly Spaced Array
• numpy.eye() in Python
• Creating a one-dimensional NumPy array
• How to create an empty and a full NumPy array?
• Create a Numpy array filled with all zeros | Python
• How to generate 2-D Gaussian array using NumPy?
• How to create a vector in Python using NumPy
• Python | Numpy fromrecords() method

NumPy Array Manipulation

• NumPy Copy and View of Array
• How to Copy NumPy array into another array?
• Appending values at the end of an NumPy array
• How to swap columns of a given NumPy array?
• Insert a new axis within a NumPy array
• numpy.hstack() in Python
• numpy.vstack() in python
• Joining NumPy Array
• Combining a one and a two-dimensional NumPy Array
• Python | Numpy np.ma.concatenate() method
• Python | Numpy dstack() method
• Splitting Arrays in NumPy
• How to compare two NumPy arrays?
• Find the union of two NumPy arrays
• Find unique rows in a NumPy array
• Python | Numpy np.unique() method
• numpy.trim_zeros() in Python

Matrix in NumPy

Matrix manipulation in python.

• numpy matrix operations | empty() function
• numpy matrix operations | zeros() function
• numpy matrix operations | ones() function
• numpy matrix operations | eye() function
• numpy matrix operations | identity() function
• Adding and Subtracting Matrices in Python
• Matrix Multiplication in NumPy
• Numpy ndarray.dot() function | Python
• NumPy | Vector Multiplication
• How to calculate dot product of two vectors in Python?
• Multiplication of two Matrices in Single line using Numpy in Python
• Python | Numpy np.eigvals() method
• How to Calculate the determinant of a matrix using NumPy?
• Python | Numpy matrix.transpose()
• Python | Numpy matrix.var()
• Compute the inverse of a matrix using NumPy

Operations on NumPy Array

• Numpy | Binary Operations
• Numpy | Mathematical Function
• Numpy | String Operations

Reshaping NumPy Array

• Reshape NumPy Array
• Python | Numpy matrix.resize()
• Python | Numpy matrix.reshape()
• NumPy Array Shape
• Change the dimension of a NumPy array
• numpy.ndarray.resize() function - Python
• Flatten a Matrix in Python using NumPy
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Indexing NumPy Array

• Basic Slicing and Advanced Indexing in NumPy
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• How to access different rows of a multidimensional NumPy array?
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Arithmetic operations on NumPyArray

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Linear Algebra in NumPy Array

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Sorting and Searching in NumPy Array

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Universal Functions

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In python matrix can be implemented as 2D list or 2D Array. Forming matrix from latter, gives the additional functionalities for performing various operations in matrix. These operations and array are defines in module “ numpy “. 

Operation on Matrix :  

• 1. add() :- This function is used to perform element wise matrix addition . 
• 2. subtract() :- This function is used to perform element wise matrix subtraction . 
• 3. divide() :- This function is used to perform element wise matrix division . 

Implementation:

• 4. multiply() :- This function is used to perform element wise matrix multiplication . 
• 5. dot() :- This function is used to compute the matrix multiplication, rather than element wise multiplication . 
• 6. sqrt() :- This function is used to compute the square root of each element of matrix. 
• 7. sum(x,axis) :- This function is used to add all the elements in matrix . Optional “axis” argument computes the column sum if axis is 0 and row sum if axis is 1 . 
• 8. “T” :- This argument is used to transpose the specified matrix. 

Using nested loops:

• Define matrices A and B.
• Get the number of rows and columns of the matrices using the len() function.
• Initialize matrices C, D, and E with zeros using nested loops or list comprehension.
• Use nested loops or list comprehension to perform the element-wise addition, subtraction, and division of matrices.
• Print the resulting matrices C, D, and E.

Time complexity: O(n^2) Space complexity: O(n^2)

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What 3 Studies Say About Ordered Matrix Assignment Expertise? There actually weren’t many studies that concluded that being a Matrix Matrix is equivalent to being an expert in it. I’ve dealt extensively with the various Matrix Matrix studies that I’ve seen online and among those that I read have gotten their data to be pretty far from what is usually called a MLE. As far as the best Matrix Matrix studies are concerned, I’d say that any of the studies are not as well drawn out as some are. So if you have high quality academic data you should probably read them first. Many of the studies are quite interesting articles but they’re really not meant for the purposes of studies.

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Responsibility Assignment Matrix: A Complete Overview

Dive into the world of Responsibility Assignment Matrix (RAMs), which helps assign roles and responsibilities and how they streamline Project Management. This comprehensive blog explains their purpose, benefits, and practical use, enabling effective role definition and accountability in project teams. Continue reading!

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This comprehensive blog aims to provide you with a complete overview of the Responsibility Assignment Matrix and its pivotal role in Project Management and organisational structure. 

Table of Content 

1) What is a Responsibility Assignment Matrix in Project Management? 

2)  Responsibility Assignment Matrix (RAM) goal in Project Management 

3) How to create a Responsibility Assignment Matrix? 

4) Benefits of Responsibility Assignment Matrix 

5) Developing Responsibility Assignment Matrix (RAM) best practices 

6) Conclusion  

What is a Responsibility Assignment Matrix in Project Management? 

A Responsibility Assignment Matrix (RAM) in project management is a tool that outlines and defines the roles and responsibilities of individuals or groups involved in a project. Its purpose is to ensure that everyone understands their specific duties and tasks. The primary purpose of a RAM is to bring clarity to the project structure, helping to prevent confusion, overlap, and accountability issues throughout the project lifecycle.

RAM in Project Management is also known as Responsible, Accountable, Consulted and Informed (RACI). RACI represents different levels of roles and responsibilities for individuals or teams:

a) Responsible: The individual or group in responsible for finishing a certain job or project. They are the ones who perform the work.

b) Accountable: The individuals who have complete responsibility and decision-making authority over the job or result. They ensure that the task is completed and of satisfactory quality.

c) Consulted: Individuals or stakeholders are consulted for their views or skills before to making a decision or taking actions. They contribute essential insights but may not be directly responsible for the task.

d) Informed: Individuals or stakeholders who need regular updates on the task’s progress or result. They are not actively involved in its conclusion, but they must be notified of any advancements.  

Responsibility Assignment Matrix goal in Project Management 

The primary goal of a Responsibility Assignment Matrix in Project Management is to clearly define and communicate the roles and responsibilities of individuals or teams involved in a project. Here are the key goals of using a RAM in Project Management: 

a) Clear roles and responsibilities: The RAM establishes clear roles and responsibilities for each team member, minimising confusion and conflicts.

b) Enhanced communication: Documenting roles and responsibilities concisely in the RAM facilitates effective communication within the project team. Also, enabling quick identification of contacts for specific issues or inquires.

c) Conflict resolution: BY operating a reference point, the RAM helps to resolve conflicts or misunderstandings about responsibilities, providing a foundation for conversation and conflict resolution.

d) Improved project control: With the RAM in place, Project Managers and stakeholders can more easily monitor project progress, identifying task accountability and monitoring work package status.

e) Efficiency and accountability: By allocating responsibility to each project aspect, the RAM promotes accountability among team members, resulting in increased efficiency as everyone understands their duties and expectations.

f) Risk Management: The clear roles of RAM help detect potential hazards associated with functional gaps or overlaps, enabling proactive risk mitigation strategies.  

g) Optimal Resource Allocation: Project managers can optimise resource allocation by understanding task ownership and workload distribution as described in the RAM.

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How to create a Responsibility Assignment Matrix? 

Identify the project scope 

Start by recognising the project's scope or the exact purpose for which you are building RAM. This includes establishing project objectives, outcomes, and critical milestones.

Define roles and responsibilities 

a) Identify the key roles involved in the project. Common roles may include Project Manager, Team Leader, Team Member, Stakeholders, and other relevant positions. 

b) Clearly define the responsibilities associated with each role. These responsibilities should be specific and measurable so there is no ambiguity. 

Choose a framework - RACI, RASCI, or DACI 

a) Select a framework for your RAM. The most commonly used frameworks are RACI, RASCI, and DACI: 

b) RACI: RACI   Stands for Responsible, Accountable, Consulted, and Informed. It defines who is responsible for a task, who is accountable for its completion, who needs to be consulted, and who needs to be informed. 

c) RASCI: RASCI Similar to RACI, but with an additional role, the "S" for Support. This framework further clarifies who provides support for a task. 

d) DACI: This framework is similar to RASCI but adds the role of Driver. The Driver is responsible for ensuring that a task is completed. 

Assign roles using the framework 

a) For each task or work package within the project, assign the relevant roles using the chosen framework. Each task should have a Responsible person, an Accountable person, and, if necessary, people who need to be Consulted, Informed, or Supported. 

b) Be specific and ensure that there is only one person designated as "Accountable" for each task to avoid confusion. 

Document the RAM 

a) Create a table or chart that lists all the tasks or work packages on one axis and the identified roles on the other. 

b) Fill in the matrix with the appropriate role designations (R, A, C, I, S, D) for each task and role. 

Review and validate 

Share the RAM with the project team and stakeholders for evaluation and approval. Confirm that everyone agrees on the roles and responsibilities.

Implement and communicate 

Once the RAM has been completed and approved, communicate it to the project team and other stakeholders. Ensure that everyone understands their jobs and responsibilities. 

Regularly update the RAM 

As the project evolves, it is critical to improve the RAM as necessary. Roles and duties might shift, and new tasks may develop. Keep the RAM current to reflect the project's evolving needs.

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Benefits of Responsibility Assignment Matrix

a) Clarify roles and responsibilities: A RAM clearly defines who is responsible for each task, which helps prevent confusion and ensures that team members understand their roles.

b) Improved communication: The RAM serves as a central reference point for roles and responsibilities, promoting effective communication within the project team. Team members can quickly identify who to contact for specific issues or questions.

c) Conflict resolution: When there is a dispute or ambiguity regarding responsibilities, the RAM provides a basis for discussion and conflict resolution. It helps identify where accountability lies and facilitates problem-solving.

d) Enhanced project control: The RAM allows Project Managers and stakeholders to monitor and manage the project's development.  It helps you to track the status of assignments and ensure that tasks are getting done as planned.

e) Efficiency and accountability: Clearly defined roles and responsibilities create a sense of responsibility among team members, maybe resulting in improved productivity. When people understand what is required of them, and they are more likely to fulfil it.

f) Resource allocation: The RAM helps Project Managers optimise resource allocation by understanding who is responsible for specific tasks. This ensures that workloads are distributed evenly and that resources are used efficiently. 

Developing Responsibility Assignment Matrix (RAM) best practices 

Developing a Responsibility Assignment Matrix (RAM) is a critical aspect of project management. To create an effective RAM, consider the following best practices: 

a) Involve key stakeholders: Define roles and responsibilities after consulting with project stakeholders, team members, and subject matter experts. Their participation can provide useful information about the project's objectives. 

b) Keep it simple: Use a simple framework (e.g., RACI, RASCI, or DACI) that team members are able to quickly understand and use.

c) Use clear and specific language: In order to prevent confusion, write tasks clearly and precisely. Avoid using confusing or unclear terms that may lead to confusion.

d) Designate a single "accountable" person: Assign only one person as "Accountable" for each task. This individual is ultimately responsible for the task's completion. Multiple accountable persons can lead to confusion and accountability issues. 

e) Consult and inform appropriately:   It is important to carefully consider the individuals who need to be consulted and informed for each task. Avoid unnecessary involvement, which can lead to inefficiency. Ensure that the right people are included in these roles. 

f) Review and validate with the team: Share the RAM with the project team and stakeholders for feedback and validation. Ensure that all parties agree with the assigned roles and responsibilities. 

g) Document assumptions and clarifications: If certain roles and responsibilities are based on assumptions or require clarification, document these notes alongside the RAM. This can help avoid misunderstandings in the future. 

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Conclusion  

A Responsibility Assignment Matrix is an important tool in Project Management and organisational systems. Its importance comes from its capacity to define, assign, and explain the roles and responsibilities, resulting in improved project effectiveness and performance.

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Frequently Asked Questions

Creating a Responsibility Matrix is appropriate for the project's initiation stage. It outlines roles, duties, and communication channels to promote responsibility throughout the project's lifecycle. 

The Responsibility Assignment Matrix or RACI model divides tasks as Responsible, Accountable, Consulted, or Informed. It promotes transparency and accountability among the project group.

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