Matrices and Matrix Operations

Learning Objectives

In this section, you will:

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

Figure 1 (credit: “SD Dirk,” Flickr)

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

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

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,𝐴,𝐵, and C𝐶 are shown below.

A=[1324],B=⎡⎣⎢1072−58762⎤⎦⎥,C=⎡⎣⎢−103321⎤⎦⎥𝐴=[1234],𝐵=[1270−56782],𝐶=[−103321]

Describing Matrices

A matrix is often referred to by its size or dimensions: m×n𝑚×𝑛 indicating m𝑚 rows and n𝑛 columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix A𝐴 identified as aij,𝑎𝑖𝑗, we look for the entry in row i,𝑖, column j.𝑗. In matrix A,  𝐴,   shown below, the entry in row 2, column 3 is a23.𝑎23.

A=⎡⎣⎢a11a21a31a12a22a32a13a23a33⎤⎦⎥𝐴=[𝑎11𝑎12𝑎13𝑎21𝑎22𝑎23𝑎31𝑎32𝑎33]

A square matrix is a matrix with dimensions n×n,𝑛×𝑛, meaning that it has the same number of rows as columns. The 3×33×3 matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions 1×n.1×𝑛.

[a11a12a13][𝑎11𝑎12𝑎13]

A column matrix is a matrix consisting of one column with dimensions m×1.𝑚×1.

⎡⎣⎢a11a21a31⎤⎦⎥[𝑎11𝑎21𝑎31]

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.

MATRICES

A matrix is a rectangular array of numbers that is usually named by a capital letter: A,B,C,𝐴,𝐵,𝐶, and so on. Each entry in a matrix is referred to as aij,𝑎𝑖𝑗, such that i𝑖 represents the row and j𝑗 represents the column. Matrices are often referred to by their dimensions: m×n𝑚×𝑛 indicating m𝑚 rows and n𝑛 columns.

EXAMPLE 1

Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix A:𝐴:

1. ⓐWhat are the dimensions of matrix A?𝐴?
2. ⓑWhat are the entries at a31𝑎31 and a22?𝑎22?A=⎡⎣⎢22314107−2⎤⎦⎥𝐴=[21024731−2]

Solution

1. ⓐThe dimensions are 3×33×3 because there are three rows and three columns.
2. ⓑEntry a31𝑎31 is the number at row 3, column 1, which is 3. The entry a22𝑎22 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.

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 3×33×3 matrix and another 3×33×3 matrix, but we cannot add or subtract a 2×32×3 matrix and a 3×33×3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

ADDING AND SUBTRACTING MATRICES

Given matrices A𝐴 and B𝐵 of like dimensions, addition and subtraction of A𝐴 and B𝐵 will produce matrix C𝐶 or
matrix D𝐷 of the same dimension.

A+B=Csuch that aij+bij=cij𝐴+𝐵=𝐶such that 𝑎𝑖𝑗+𝑏𝑖𝑗=𝑐𝑖𝑗

A−B=Dsuch that aij−bij=dij𝐴−𝐵=𝐷such that 𝑎𝑖𝑗−𝑏𝑖𝑗=𝑑𝑖𝑗

Matrix addition is commutative.

A+B=B+A𝐴+𝐵=𝐵+𝐴

It is also associative.

(A+B)+C=A+(B+C)(𝐴+𝐵)+𝐶=𝐴+(𝐵+𝐶)

EXAMPLE 2

Finding the Sum of Matrices

Find the sum of A𝐴 and B,𝐵, given

A=[acbd]  and  B=[egfh]𝐴=[𝑎𝑏𝑐𝑑]  and  𝐵=[𝑒𝑓𝑔ℎ]

Solution

Add corresponding entries.

A+B=[acbd]+[egfh]        =[a+ec+gb+fd+h]𝐴+𝐵=[𝑎𝑏𝑐𝑑]+[𝑒𝑓𝑔ℎ]        =[𝑎+𝑒𝑏+𝑓𝑐+𝑔𝑑+ℎ]

EXAMPLE 3

Adding Matrix A and Matrix B

Find the sum of A𝐴 and B.𝐵.

A=[4312] and  B=[5097]𝐴=[4132] and  𝐵=[5907]

Solution

Add corresponding entries. Add the entry in row 1, column 1, a11,𝑎11, of matrix A𝐴 to the entry in row 1, column 1, b11,𝑏11, of B.𝐵. Continue the pattern until all entries have been added.

A+B=[4312]+[5097]        =[4+53+01+92+7]        =[93109]𝐴+𝐵=[4132]+[5907]        =[4+51+93+02+7]        =[91039]

EXAMPLE 4

Finding the Difference of Two Matrices

Find the difference of A𝐴 and B.𝐵.

A=[−2031] and  B=[8514]𝐴=[−2301] and  𝐵=[8154]

Solution

We subtract the corresponding entries of each matrix.

A−B=[−2031]−[8514]        =[−2−80−53−11−4]        =[−10−52−3]𝐴−𝐵=[−2301]−[8154]        =[−2−83−10−51−4]        =[−102−5−3]

EXAMPLE 5

Finding the Sum and Difference of Two 3 x 3 Matrices

Given A𝐴 and B:𝐵:

1. ⓐFind the sum.
2. ⓑFind the difference.

A=⎡⎣⎢2144−1012−2−2102⎤⎦⎥and B=⎡⎣⎢60−510−122−2−4−2⎤⎦⎥𝐴=[2−10−21412104−22]and 𝐵=[610−20−12−4−52−2]

Solution

1. ⓐAdd the corresponding entries.A+B=⎡⎣⎢2144−1012−2−2102⎤⎦⎥+⎡⎣⎢60−510−122−2−4−2⎤⎦⎥=⎡⎣⎢2+614+04−5−10+1012−12−2+2−2−210−42−2⎤⎦⎥=⎡⎣⎢814−1000−460⎤⎦⎥𝐴+𝐵=[2−10−21412104−22]+[610−20−12−4−52−2]=[2+6−10+10−2−214+012−1210−44−5−2+22−2]=[80−41406−100]
2. ⓑSubtract the corresponding entries.A−B=⎡⎣⎢2144−1012−2−2102⎤⎦⎥−⎡⎣⎢60−510−122−2−4−2⎤⎦⎥=⎡⎣⎢2−614−04+5−10−1012+12−2−2−2+210+42+2⎤⎦⎥=⎡⎣⎢−4149−2024−40144⎤⎦⎥𝐴−𝐵=[2−10−21412104−22]−[610−20−12−4−52−2]=[2−6−10−10−2+214−012+1210+44+5−2−22+2]=[−4−2001424149−44]

TRY IT #1

Add matrix A𝐴 and matrix B.𝐵.

A=⎡⎣⎢21160−3⎤⎦⎥ and  B=⎡⎣⎢31−4−253⎤⎦⎥𝐴=[26101−3] and  𝐵=[3−215−43]

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 2.

	Lab A	Lab B
Computers	15	27
Computer Tables	16	34
Chairs	16	34

Converting the data to a matrix, we have

C2013=⎡⎣⎢151616273434⎤⎦⎥𝐶2013=[151616273434]

To calculate how much computer equipment will be needed, we multiply all entries in matrix C𝐶 by 0.15.

(0.15)C2013=⎡⎣⎢(0.15)15(0.15)16(0.15)16(0.15)27(0.15)34(0.15)34⎤⎦⎥=⎡⎣⎢2.252.42.44.055.15.1⎤⎦⎥(0.15)𝐶2013=[(0.15)15(0.15)16(0.15)16(0.15)27(0.15)34(0.15)34]=[2.252.42.44.055.15.1]

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

⎡⎣⎢333566⎤⎦⎥[333566]

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

⎡⎣⎢151616273434⎤⎦⎥+⎡⎣⎢333566⎤⎦⎥=⎡⎣⎢181919324040⎤⎦⎥[151616273434]+[333566]=[181919324040]

This means

C2014=⎡⎣⎢181919324040⎤⎦⎥𝐶2014=[181919324040]

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=[a11a21a12a22]𝐴=[𝑎11𝑎12𝑎21𝑎22]

the scalar multiple cA𝑐𝐴 is

cA=c[a11a21a12a22]   =[ca11ca21ca12ca22]𝑐𝐴=𝑐[𝑎11𝑎12𝑎21𝑎22]   =[𝑐𝑎11𝑐𝑎12𝑐𝑎21𝑐𝑎22]

Scalar multiplication is distributive. For the matrices A,B,𝐴,𝐵, and C𝐶 with scalars a𝑎 and b,𝑏,

a(A+B)=aA+aB(a+b)A=aA+bA𝑎(𝐴+𝐵)=𝑎𝐴+𝑎𝐵(𝑎+𝑏)𝐴=𝑎𝐴+𝑏𝐴

EXAMPLE 6

Multiplying the Matrix by a Scalar

Multiply matrix A𝐴 by the scalar 3.

A=[8514]𝐴=[8154]

Solution

Multiply each entry in A𝐴 by the scalar 3.

3A=3[8514]= [3⋅83⋅53⋅13⋅4]= [2415312]3𝐴=3[8154]= [3⋅83⋅13⋅53⋅4]= [2431512]

TRY IT #2

Given matrix B,𝐵, find −2B−2𝐵 where

B=[4312]𝐵=[4132]

EXAMPLE 7

Finding the Sum of Scalar Multiples

Find the sum 3A+2B.3𝐴+2𝐵.

A=⎡⎣⎢104−2−1302−6⎤⎦⎥and B=⎡⎣⎢−1002−3112−4⎤⎦⎥𝐴=[1−200−1243−6]and 𝐵=[−1210−3201−4]

Solution

First, find 3A,3𝐴, then 2B.2𝐵.

3A=⎡⎣⎢3⋅13⋅03⋅43(−2)3(−1)3⋅33⋅03⋅23(−6)⎤⎦⎥=⎡⎣⎢3012−6−3906−18⎤⎦⎥3𝐴=[3⋅13(−2)3⋅03⋅03(−1)3⋅23⋅43⋅33(−6)]=[3−600−36129−18]

2B=⎡⎣⎢2(−1)2⋅02⋅02⋅22(−3)2⋅12⋅12⋅22(−4)⎤⎦⎥=⎡⎣⎢−2004−6224−8⎤⎦⎥2𝐵=[2(−1)2⋅22⋅12⋅02(−3)2⋅22⋅02⋅12(−4)]=[−2420−6402−8]

Now, add 3A+2B.3𝐴+2𝐵.

3A+2B=⎡⎣⎢3012−6−3906−18⎤⎦⎥+⎡⎣⎢−2004−6224−8⎤⎦⎥            =⎡⎣⎢3−20+012+0−6+4−3−69+20+26+4−18−8⎤⎦⎥            =⎡⎣⎢1012−2−911210−26⎤⎦⎥3𝐴+2𝐵=[3−600−36129−18]+[−2420−6402−8]            =[3−2−6+40+20+0−3−66+412+09+2−18−8]            =[1−220−9101211−26]

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 A𝐴 is an m×r𝑚×𝑟 matrix and B𝐵 is an r×n𝑟×𝑛 matrix, then the product matrix AB𝐴𝐵 is an m×n𝑚×𝑛 matrix. For example, the product AB𝐴𝐵 is possible because the number of columns in A𝐴 is the same as the number of rows in B.𝐵. If the inner dimensions do not match, the product is not defined.

We multiply entries of A𝐴 with entries of B𝐵 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 i𝑖 of AB,𝐴𝐵, we multiply the entries in row i𝑖 of A𝐴 by column j𝑗 in B𝐵 and add. For example, given matrices A𝐴 and B,𝐵, where the dimensions of A𝐴 are 2×32×3 and the dimensions of B𝐵 are 3×3,3×3, the product of AB𝐴𝐵 will be a 2×32×3 matrix.

A=[a11a21a12a22a13a23]and B=⎡⎣⎢b11b21b31b12b22b32b13b23b33⎤⎦⎥𝐴=[𝑎11𝑎12𝑎13𝑎21𝑎22𝑎23]and 𝐵=[𝑏11𝑏12𝑏13𝑏21𝑏22𝑏23𝑏31𝑏32𝑏33]

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

1. To obtain the entry in row 1, column 1 of AB,𝐴𝐵, multiply the first row in A𝐴 by the first column in B,𝐵, and add.[a11a12a13]⎡⎣⎢b11b21b31⎤⎦⎥=a11⋅b11+a12⋅b21+a13⋅b31[𝑎11𝑎12𝑎13][𝑏11𝑏21𝑏31]=𝑎11⋅𝑏11+𝑎12⋅𝑏21+𝑎13⋅𝑏31
2. To obtain the entry in row 1, column 2 of AB,𝐴𝐵, multiply the first row of A𝐴 by the second column in B,𝐵, and add.[a11a12a13]⎡⎣⎢b12b22b32⎤⎦⎥=a11⋅b12+a12⋅b22+a13⋅b32[𝑎11𝑎12𝑎13][𝑏12𝑏22𝑏32]=𝑎11⋅𝑏12+𝑎12⋅𝑏22+𝑎13⋅𝑏32
3. To obtain the entry in row 1, column 3 of AB,𝐴𝐵, multiply the first row of A𝐴 by the third column in B,𝐵, and add.[a11a12a13]⎡⎣⎢b13b23b33⎤⎦⎥=a11⋅b13+a12⋅b23+a13⋅b33[𝑎11𝑎12𝑎13][𝑏13𝑏23𝑏33]=𝑎11⋅𝑏13+𝑎12⋅𝑏23+𝑎13⋅𝑏33

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

AB=⎡⎣⎢a11⋅b11+a12⋅b21+a13⋅b31a21⋅b11+a22⋅b21+a23⋅b31a11⋅b12+a12⋅b22+a13⋅b32a21⋅b12+a22⋅b22+a23⋅b32a11⋅b13+a12⋅b23+a13⋅b33a21⋅b13+a22⋅b23+a23⋅b33⎤⎦⎥𝐴𝐵=[𝑎11⋅𝑏11+𝑎12⋅𝑏21+𝑎13⋅𝑏31𝑎21⋅𝑏11+𝑎22⋅𝑏21+𝑎23⋅𝑏31𝑎11⋅𝑏12+𝑎12⋅𝑏22+𝑎13⋅𝑏32𝑎21⋅𝑏12+𝑎22⋅𝑏22+𝑎23⋅𝑏32𝑎11⋅𝑏13+𝑎12⋅𝑏23+𝑎13⋅𝑏33𝑎21⋅𝑏13+𝑎22⋅𝑏23+𝑎23⋅𝑏33]

PROPERTIES OF MATRIX MULTIPLICATION

For the matrices A,B,𝐴,𝐵, and C𝐶 the following properties hold.

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

Note that matrix multiplication is not commutative.

EXAMPLE 8

Multiplying Two Matrices

Multiply matrix A𝐴 and matrix B.𝐵.

A=[1324] and  B=[5768]𝐴=[1234] and  𝐵=[5678]

Solution

First, we check the dimensions of the matrices. Matrix A𝐴 has dimensions 2×22×2 and matrix B𝐵 has dimensions 2×2.2×2. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions 2×2.2×2.

We perform the operations outlined previously.

EXAMPLE 9

Multiplying Two Matrices

Given A𝐴 and B:𝐵:

1. ⓐ Find AB.𝐴𝐵.
2. ⓑ Find BA.𝐵𝐴.

A=[−123405]and  B=⎡⎣⎢5−42−103⎤⎦⎥𝐴=[−123405]and  𝐵=[5−42−103]

Solution

1. ⓐAs the dimensions of A𝐴 are 2×32×3 and the dimensions of B𝐵 are 3×2,3×2, these matrices can be multiplied together because the number of columns in A𝐴 matches the number of rows in B.𝐵. The resulting product will be a 2×22×2 matrix, the number of rows in A𝐴 by the number of columns in B.𝐵.AB=[−142035]  ⎡⎣⎢5−42−103⎤⎦⎥=[−1(5)+2(−4)+3(2)4(5)+0(−4)+5(2)−1(−1)+2(0)+3(3)4(−1)+0(0)+5(3)]=[−7301011]𝐴𝐵=[−123405]  [5−1−4023]=[−1(5)+2(−4)+3(2)−1(−1)+2(0)+3(3)4(5)+0(−4)+5(2)4(−1)+0(0)+5(3)]=[−7103011]
2. ⓑThe dimensions of B𝐵 are 3×23×2 and the dimensions of A𝐴 are 2×3.2×3. The inner dimensions match so the product is defined and will be a 3×33×3 matrix.BA=⎡⎣⎢5−42−103⎤⎦⎥  [−142035]=⎡⎣⎢5(−1)+−1(4)−4(−1)+0(4)2(−1)+3(4)5(2)+−1(0)−4(2)+0(0)2(2)+3(0)5(3)+−1(5)−4(3)+0(5)2(3)+3(5)⎤⎦⎥=⎡⎣⎢−941010−8410−1221⎤⎦⎥𝐵𝐴=[5−1−4023]  [−123405]=[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)]=[−910104−8−1210421]

Analysis

Notice that the products AB𝐴𝐵 and BA𝐵𝐴 are not equal.

AB=[−7301011]≠⎡⎣⎢−941010−8410−1221⎤⎦⎥=BA𝐴𝐵=[−7103011]≠[−910104−8−1210421]=𝐵𝐴

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 3×43×4 and matrix B with dimension 4×2.4×2. 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 10

Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have Table 3, 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 Table 4.

Goal	$300
Ball	$10
Jersey	$30

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

E=⎡⎣⎢63014102420⎤⎦⎥𝐸=[63014102420]

The cost matrix is written as

C=[3001030]𝐶=[3001030]

We perform matrix multiplication to obtain costs for the equipment.

CE=[3001030]⎡⎣⎢63014102420⎤⎦⎥=[300(6)+10(30)+30(14)300(10)+10(24)+30(20)]=[2,5203,840]𝐶𝐸=[3001030][61030241420]=[300(6)+10(30)+30(14)300(10)+10(24)+30(20)]=[2,5203,840]

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 [A],[B],[C],…[𝐴],[𝐵],[𝐶],…
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 11

Using a Calculator to Perform Matrix Operations

Find AB−C𝐴𝐵−𝐶 given

A=⎡⎣⎢−15411025−73432−28−2⎤⎦⎥,B=⎡⎣⎢45−2462152−48−3719−31⎤⎦⎥,and C=⎡⎣⎢−10025−67−89−5642−9874−75⎤⎦⎥.𝐴=[−15253241−7−281034−2],𝐵=[4521−37−2452196−48−31],and 𝐶=[−100−89−9825−5674−6742−75].

Solution

On the matrix page of the calculator, we enter matrix A𝐴 above as the matrix variable [A],[𝐴], matrix B𝐵 above as the matrix variable [B],[𝐵], and matrix C𝐶 above as the matrix variable [C].[𝐶].

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

[A][B]−[C][𝐴][𝐵]−[𝐶]

The calculator gives us the following matrix.

⎡⎣⎢−9831,820−311−4621,8972,032136−856413⎤⎦⎥[−983−4621361,8201,897−856−3112,032413]