## Matrices and Matrix Operations

### Learning Outcomes

• Determine the dimensions of a matrix.
• Add and subtract two matrices.
• Multiply a matrix by a scalar, sum scalar multiples of matrices.
• Multiply two matrices together.
• Use a calculator to perform operations on matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. The table shows the needs of both teams.

Wildcats Mud Cats
Goals 6 10
Balls 30 24
Jerseys 14 20

Two teams competing in a soccer game. (credit: “SD Dirk,” Flickr)

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. ## Adding and Subtracting 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 $A,B,\text{}$ and $C$ are shown below. $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 7\\ 0& -5& 6\\ 7& 8& 2\end{array}\right],C=\left[\begin{array}{c}-1\\ 0\\ 3\end{array}\begin{array}{c}3\\ 2\\ 1\end{array}\right]$ ### Describing Matrices A matrix is often referred to by its size or dimensions: $\text{ }m\text{ }\times \text{ }n\text{ }$ 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 ${a}_{ij},\text{}$ we look for the entry in row $i,\text{}$ column $j$. In matrix $A$ shown below, the entry in row 2, column 3 is ${a}_{23}$. $A=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]$ A square matrix is a matrix with dimensions $\text{ }n\text{ }\times \text{ }n,\text{}$ meaning that it has the same number of rows as columns. The $3\times 3$ matrix above is an example of a square matrix. A row matrix is a matrix consisting of one row with dimensions $1\text{ }\times \text{ }n$. $\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]$ A column matrix is a matrix consisting of one column with dimensions $m\text{ }\times \text{ }1$. $\left[\begin{array}{c}{a}_{11}\\ {a}_{21}\\ {a}_{31}\end{array}\right]$ 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. ### A General Note: Matrices A matrix is a rectangular array of numbers that is usually named by a capital letter: $A,B,C,\text{}$ 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: $m\times n$ indicating $m$ rows and $n$ columns. ### Example: Finding the Dimensions of the Given Matrix and Locating Entries Given matrix $A:$ 1. What are the dimensions of matrix $A?$ 2. What are the entries at ${a}_{31}$ and ${a}_{22}?$ $A=\left[\begin{array}{rrrr}\hfill 2& \hfill & \hfill 1& \hfill 0\\ \hfill 2& \hfill & \hfill 4& \hfill 7\\ \hfill 3& \hfill & \hfill 1& \hfill -2\end{array}\right]$ ### Try it ### 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. In order 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 $\text{ }3\text{ }\times \text{ }3\text{ }$ matrix and another $\text{ }3\text{ }\times \text{ }3\text{ }$ matrix, but we cannot add or subtract a $\text{ }2\text{ }\times \text{ }3\text{ }$ matrix and a $\text{ }3\text{ }\times \text{ }3\text{ }$ matrix because some entries in one matrix will not have a corresponding entry in the other matrix. ### A General Note: 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. $A+B=C\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}$ $A-B=D\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}$ Matrix addition is commutative. $A+B=B+A$ It is also associative. $\left(A+B\right)+C=A+\left(B+C\right)$ ### Example: Finding the Sum of Matrices Find the sum of $A$ and $B \text{}$ given $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{ and }B=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]$ ### Example: Adding Matrix A and Matrix B Find the sum of $A$ and $B$. $A=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]\text{ and }B=\left[\begin{array}{cc}5& 9\\ 0& 7\end{array}\right]$ ### Example: Finding the Difference of Two Matrices Find the difference of $A$ and $B$. $A=\left[\begin{array}{cc}-2& 3\\ 0& 1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}8& 1\\ 5& 4\end{array}\right]$ ### Example: Finding the Sum and Difference of Two 3 x 3 Matrices Given $A$ and $B:$ 1. Find the sum. 2. Find the difference. $A=\left[\begin{array}{rrr}\hfill 2& \hfill -10& \hfill -2\\ \hfill 14& \hfill 12& \hfill 10\\ \hfill 4& \hfill -2& \hfill 2\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill 6& \hfill 10& \hfill -2\\ \hfill 0& \hfill -12& \hfill -4\\ \hfill -5& \hfill 2& \hfill -2\end{array}\right]$ ### Try It Add matrix $A$ and matrix $B$. $A=\left[\begin{array}{rr}\hfill 2& \hfill 6\\ \hfill 1& \hfill 0\\ \hfill 1& \hfill -3\end{array}\right]\text{ and }B=\left[\begin{array}{rr}\hfill 3& \hfill -2\\ \hfill 1& \hfill 5\\ \hfill -4& \hfill 3\end{array}\right]$ ## Products of Matrices 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 the table below. Lab A Lab B Computers 15 27 Computer Tables 16 34 Chairs 16 34 Converting the data to a matrix, we have ${C}_{2013}=\left[\begin{array}{c}15\\ 16\\ 16\end{array}\begin{array}{c}27\\ 34\\ 34\end{array}\right]$ To calculate how much computer equipment will be needed, we multiply all entries in matrix $C$ by 0.15. $\left(0.15\right){C}_{2013}=\left[\begin{array}{c}\left(0.15\right)15\\ \left(0.15\right)16\\ \left(0.15\right)16\end{array}\begin{array}{c}\left(0.15\right)27\\ \left(0.15\right)34\\ \left(0.15\right)34\end{array}\right]=\left[\begin{array}{c}2.25\\ 2.4\\ 2.4\end{array}\begin{array}{c}4.05\\ 5.1\\ 5.1\end{array}\right]$ We must round up to the next integer, so the amount of new equipment needed is $\left[\begin{array}{c}3\\ 3\\ 3\end{array}\begin{array}{c}5\\ 6\\ 6\end{array}\right]$ Adding the two matrices as shown below, we see the new inventory amounts. $\left[\begin{array}{c}15\\ 16\\ 16\end{array}\begin{array}{c}27\\ 34\\ 34\end{array}\right]+\left[\begin{array}{c}3\\ 3\\ 3\end{array}\begin{array}{c}5\\ 6\\ 6\end{array}\right]=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\begin{array}{c}32\\ 40\\ 40\end{array}\right]$ This means ${C}_{2014}=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\begin{array}{c}32\\ 40\\ 40\end{array}\right]$ 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. ### A General Note: Scalar Multiplication Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given $A=\left[\begin{array}{cccc}{a}_{11}& & & {a}_{12}\\ {a}_{21}& & & {a}_{22}\end{array}\right]$ the scalar multiple $cA$ is $\begin{array}{ll}cA & =c\left[\begin{array}{ccc}{a}_{11}& & {a}_{12}\\ {a}_{21}& & {a}_{22}\end{array}\right]\hfill \\ & =\left[\begin{array}{ccc}c{a}_{11}& & c{a}_{12}\\ c{a}_{21}& & c{a}_{22}\end{array}\right]\hfill \end{array}$ Scalar multiplication is distributive. For the matrices $A,B$, and $C$ with scalars $a$ and $b$, $\begin{array}{l}\\ \begin{array}{c}a\left(A+B\right)=aA+aB\\ \left(a+b\right)A=aA+bA\end{array}\end{array}$ ### Example: Multiplying the Matrix by a Scalar Multiply matrix $A$ by the scalar 3. $A=\left[\begin{array}{cc}8& 1\\ 5& 4\end{array}\right]$ ### Try It Given matrix $B,\text{}$ find $-2B$ where $B=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]$ ### Example: Finding the Sum of Scalar Multiples Find the sum $3A+2B$. $A=\left[\begin{array}{rrr}\hfill 1& \hfill -2& \hfill 0\\ \hfill 0& \hfill -1& \hfill 2\\ \hfill 4& \hfill 3& \hfill -6\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill -1& \hfill 2& \hfill 1\\ \hfill 0& \hfill -3& \hfill 2\\ \hfill 0& \hfill 1& \hfill -4\end{array}\right]$ ### Try it ### 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 $\text{ }m\text{ }\times \text{ }r\text{ }$ matrix and $B$ is an $\text{ }r\text{ }\times \text{ }n\text{ }$ matrix, then the product matrix $AB$ is an $\text{ }m\text{ }\times \text{ }n\text{ }$ 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 $AB,\text{}$ we multiply the entries in row $i$ of $A$ by column $j$ in $B$ and add. For example, given matrices $A$ and $B,\text{}$ where the dimensions of $A$ are $2\text{ }\times \text{ }3$ and the dimensions of $B$ are $3\text{ }\times \text{ }3,\text{}$ the product of $AB$ will be a $2\text{ }\times \text{ }3$ matrix. $A=\left[\begin{array}{rrr}\hfill {a}_{11}& \hfill {a}_{12}& \hfill {a}_{13}\\ \hfill {a}_{21}& \hfill {a}_{22}& \hfill {a}_{23}\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill {b}_{11}& \hfill {b}_{12}& \hfill {b}_{13}\\ \hfill {b}_{21}& \hfill {b}_{22}& \hfill {b}_{23}\\ \hfill {b}_{31}& \hfill {b}_{32}& \hfill {b}_{33}\end{array}\right]$ Multiply and add as follows to obtain the first entry of the product matrix $AB$. 1. To obtain the entry in row 1, column 1 of $AB,\text{}$ multiply the first row in $A$ by the first column in $B$ and add. $\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{11}\\ {b}_{21}\\ {b}_{31}\end{array}\right]={a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}$ 2. To obtain the entry in row 1, column 2 of $AB,\text{}$ multiply the first row of $A$ by the second column in $B$ and add. $\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{12}\\ {b}_{22}\\ {b}_{32}\end{array}\right]={a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}$ 3. To obtain the entry in row 1, column 3 of $AB,\text{}$ multiply the first row of $A$ by the third column in $B$ and add. $\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{13}\\ {b}_{23}\\ {b}_{33}\end{array}\right]={a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}$ 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 $AB=\left[\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}\\ \end{array}\\ {a}_{21}\cdot {b}_{11}+{a}_{22}\cdot {b}_{21}+{a}_{23}\cdot {b}_{31}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}\\ \end{array}\\ {a}_{21}\cdot {b}_{12}+{a}_{22}\cdot {b}_{22}+{a}_{23}\cdot {b}_{32}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}\\ \end{array}\\ {a}_{21}\cdot {b}_{13}+{a}_{22}\cdot {b}_{23}+{a}_{23}\cdot {b}_{33}\end{array}\right]$ ### A General Note: Properties of Matrix Multiplication For the matrices $A,B,\text{}$ and $C$ the following properties hold. • Matrix multiplication is associative: $\left(AB\right)C=A\left(BC\right)$ • Matrix multiplication is distributive: $\begin{array}{l}\begin{array}{l}\\ C\left(A+B\right)=CA+CB,\end{array}\hfill \\ \left(A+B\right)C=AC+BC.\hfill \end{array}$ Note that matrix multiplication is not commutative. ### Example: Multiplying Two Matrices Multiply matrix $A$ and matrix $B$. $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\text{ and }B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$ ### Example: Multiplying Two Matrices Given $A$ and $B:$ 1. Find $AB$. 2. Find $BA$. $A=\left[\begin{array}{l}\begin{array}{ccc}-1& 2& 3\end{array}\hfill \\ \begin{array}{ccc}4& 0& 5\end{array}\hfill \end{array}\right]\text{ and }B=\left[\begin{array}{c}5\\ -4\\ 2\end{array}\begin{array}{c}-1\\ 0\\ 3\end{array}\right]$ ### Q & A Is it possible for AB to be defined but not BA? Yes, consider a matrix A with dimension $3\times 4$ and matrix B with dimension $4\times 2$. 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: Using Matrices in Real-World Problems Let’s return to the problem presented at the opening of this section. We have the table below, representing the equipment needs of two soccer teams. Wildcats Mud Cats Goals 6 10 Balls 30 24 Jerseys 14 20 We are also given the prices of the equipment, as shown in the table below.  Goal300 Ball $10 Jersey$30

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

$E=\left[\begin{array}{c}6\\ 30\\ 14\end{array}\begin{array}{c}10\\ 24\\ 20\end{array}\right]$

The cost matrix is written as

$C=\left[\begin{array}{ccc}300& 10& 30\end{array}\right]$
We perform matrix multiplication to obtain costs for the equipment.
$\begin{array}{l}\hfill \\ \hfill \\ CE & =\left[\begin{array}{rrr}\hfill 300& \hfill 10& \hfill 30\end{array}\right]\cdot \left[\begin{array}{rr}\hfill 6& \hfill 10\\ \hfill 30& \hfill 24\\ \hfill 14& \hfill 20\end{array}\right]\hfill \\ & =\left[\begin{array}{rr}\hfill 300\left(6\right)+10\left(30\right)+30\left(14\right)& \hfill 300\left(10\right)+10\left(24\right)+30\left(20\right)\end{array}\right]\hfill \\ & =\left[\begin{array}{rr}\hfill 2,520& \hfill 3,840\end{array}\right]\hfill \end{array}$

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

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

### Example: Using a Calculator to Perform Matrix Operations

Find $AB-C$ given

$A=\left[\begin{array}{rrr}\hfill -15& \hfill 25& \hfill 32\\ \hfill 41& \hfill -7& \hfill -28\\ \hfill 10& \hfill 34& \hfill -2\end{array}\right],B=\left[\begin{array}{rrr}\hfill 45& \hfill 21& \hfill -37\\ \hfill -24& \hfill 52& \hfill 19\\ \hfill 6& \hfill -48& \hfill -31\end{array}\right],\text{and }C=\left[\begin{array}{rrr}\hfill -100& \hfill -89& \hfill -98\\ \hfill 25& \hfill -56& \hfill 74\\ \hfill -67& \hfill 42& \hfill -75\end{array}\right]$.

## 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 $3\times 2$ matrix has three rows and two columns.
• We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix.
• Scalar multiplication involves multiplying each entry in a matrix by a constant.
• Scalar multiplication is often required before addition or subtraction can occur.
• 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 $B,\text{}$ and so on.
• Many real-world problems can often be solved using matrices.
• We can use a calculator to perform matrix operations after saving each matrix as a matrix variable.

## Glossary

column
a set of numbers aligned vertically in a matrix
entry
an element, coefficient, or constant in a matrix
matrix
a rectangular array of numbers
row
a set of numbers aligned horizontally in a matrix
scalar multiple
an entry of a matrix that has been multiplied by a scalar