<< Chapter < Page Chapter >> Page >
   a x = b    ( 1 a ) a x = ( 1 a ) b ( a −1     ) a x = ( a −1 ) b [ ( a −1 ) a ] x = ( a −1 ) b              1 x = ( a −1 ) b                x = ( a −1 ) b

The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.

We will investigate this idea in detail, but it is helpful to begin with a 2 × 2 system and then move on to a 3 × 3 system.

Solving a system of equations using the inverse of a matrix

Given a system of equations, write the coefficient matrix A , the variable matrix X , and the constant matrix B . Then

A X = B

Multiply both sides by the inverse of A to obtain the solution.

( A −1 ) A X = ( A −1 ) B [ ( A −1 ) A ] X = ( A −1 ) B I X = ( A −1 ) B X = ( A −1 ) B

If the coefficient matrix does not have an inverse, does that mean the system has no solution?

No, if the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.

Solving a 2 × 2 system using the inverse of a matrix

Solve the given system of equations using the inverse of a matrix.

3 x + 8 y = 5 4 x + 11 y = 7

Write the system in terms of a coefficient matrix, a variable matrix, and a constant matrix.

A = [ 3 8 4 11 ] , X = [ x y ] , B = [ 5 7 ]

Then

[ 3 8 4 11 ]     [ x y ] = [ 5 7 ]

First, we need to calculate A −1 . Using the formula to calculate the inverse of a 2 by 2 matrix, we have:

A −1 = 1 a d b c [ d b c a ]        = 1 3 ( 11 ) −8 ( 4 ) [ 11 −8 −4 3 ]        = 1 1 [ 11 −8 −4 3 ]

So,

A −1 = [ 11 −8 −4 3 ]

Now we are ready to solve. Multiply both sides of the equation by A −1 .

( A −1 ) A X = ( A −1 ) B [ 11 −8 −4 3 ]     [ 3 8 4 11 ]     [ x y ] = [ 11 −8 −4 3 ]     [ 5 7 ] [ 1 0 0 1 ]     [ x y ] = [ 11 ( 5 ) + ( −8 ) 7 −4 ( 5 ) + 3 ( 7 ) ] [ x y ] = [ −1 1 ]

The solution is ( −1 , 1 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Can we solve for X by finding the product B A −1 ?

No, recall that matrix multiplication is not commutative, so A −1 B B A −1 . Consider our steps for solving the matrix equation.

( A −1 ) A X = ( A −1 ) B [ ( A −1 ) A ] X = ( A −1 ) B I X = ( A −1 ) B X = ( A −1 ) B

Notice in the first step we multiplied both sides of the equation by A −1 , but the A −1 was to the left of A on the left side and to the left of B on the right side. Because matrix multiplication is not commutative, order matters.

Solving a 3 × 3 system using the inverse of a matrix

Solve the following system using the inverse of a matrix.

5 x + 15 y + 56 z = 35 −4 x −11 y −41 z = −26 x −3 y −11 z = −7

Write the equation A X = B .

[ 5 15 56 −4 −11 −41 −1 −3 −11 ]     [ x y z ] = [ 35 −26 −7 ]

First, we will find the inverse of A by augmenting with the identity.

[ 5 15 56 −4 −11 −41 −1 −3 −11 | 1 0 0 0 1 0 0 0 1 ]

Multiply row 1 by 1 5 .

[ 1 3 56 5 −4 −11 −41 −1 −3 −11 | 1 5 0 0 0 1 0 0 0 1 ]

Multiply row 1 by 4 and add to row 2.

[ 1 3 56 5 0 1 19 5 −1 −3 −11 | 1 5 0 0 4 5 1 0 0 0 1 ]

Add row 1 to row 3.

[ 1 3 56 5 0 1 19 5 0 0 1 5 | 1 5 0 0 4 5 1 0 1 5 0 1 ]

Multiply row 2 by −3 and add to row 1.

[ 1 0 1 5 0 1 19 5 0 0 1 5 | 11 5 −3 0 4 5 1 0 1 5 0 1 ]

Multiply row 3 by 5.

[ 1 0 1 5 0 1 19 5 0 0 1 | 11 5 −3 0 4 5 1 0 1 0 5 ]

Multiply row 3 by 1 5 and add to row 1.

[ 1 0 0 0 1 19 5 0 0 1 | −2 −3 1 4 5 1 0 1 0 5 ]

Multiply row 3 by 19 5 and add to row 2.

[ 1 0 0 0 1 0 0 0 1 | −2 −3 1 −3 1 −19 1 0 5 ]

So,

A −1 = [ −2 −3 1 −3 1 −19 1 0 5 ]

Multiply both sides of the equation by A −1 . We want A −1 A X = A −1 B :

[ −2 −3 1 −3 1 −19 1 0 5 ]     [ 5 15 56 −4 −11 −41 −1 −3 −11 ]     [ x y z ] = [ −2 −3 1 −3 1 −19 1 0 5 ]     [ 35 −26 −7 ]

Thus,

A −1 B = [ −70 + 78 −7 −105 −26 + 133 35 + 0 −35 ] = [ 1 2 0 ]

The solution is ( 1 , 2 , 0 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
Practice Key Terms 2

Get the best College algebra course in your pocket!





Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask