Prentice Hall Algebra 2

Chapter 12 Matrices > 12-4 Inverse Matrices and Systems

12-4 Inverse Matrices and Systems

Objective

To solve systems of equations using matrix inverses and multiplication

A solve it problem with Tyler.
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In Chapter 3, you used row operations on a matrix to solve a system of equations. Now you will solve systems by solving a matrix equation.

Essential Understanding You can solve some matrix equations eh x equals b by multiplying each side of the equation by eh super negative 1 end super , comma the inverse of matrix A.

If matrix A has an inverse, you can use it to solve the matrix equation eh x equals b . Multiply each side of the equation by eh super negative 1 end super to find X.

A X = B A − 1 A X = A − 1 B Multiply each side by A − 1 . I X = A − 1 B A − 1 A = I , the identity matrix . X = A − 1 B I X = X table with 4 rows and 3 columns , row1 column 1 , eh x , column 2 equals b , column 3 , row2 column 1 , eh super negative 1 end super eh x , column 2 equals , eh super negative 1 end super b , column 3 cap multiplyeachsideby . eh super negative 1 end super , . , row3 column 1 , i x , column 2 equals , eh super negative 1 end super b , column 3 eh super negative 1 end super eh equals i comma . theidentitymatrix . . , row4 column 1 , x , column 2 equals , eh super negative 1 end super b , column 3 i x equals x , end table

Problem 1 Solving a Matrix Equation Using an Inverse Matrix

What is the solution of each matrix equation?

[ 5 3 3 2 ] X = [ 1 − 3 ] . matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 3 , row2 column 1 , 3 , column 2 2 , end matrix x equals . matrix with 2 rows and 1 column , row1 column 1 , 1 , row2 column 1 , negative 3 , end matrix

Think

How do you know the equation has a solution?

Check det A. If det eh not equal to 0 comma you can solve the equation.
- Step 1 Evaluate det A and find A − 1 . eh super negative 1 end super , .
  
  det A = ( 5 ) ( 2 ) − ( 3 ) ( 3 ) = 1 eh equals open 5 close open 2 close minus open 3 close open 3 close equals 1
  
  A − 1 = 1 det A [ d − b − c a ] = 1 1 [ 2 − 3 − 3 5 ] = [ 2 − 3 − 3 5 ] eh super negative 1 end super , equals . fraction 1 , over det eh end fraction . matrix with 2 rows and 2 columns , row1 column 1 , d , column 2 negative b , row2 column 1 , negative c , column 2 eh , end matrix . equals , 1 over 1 . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 3 , row2 column 1 , negative 3 , column 2 5 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 3 , row2 column 1 , negative 3 , column 2 5 , end matrix

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12-4 Inverse Matrices and Systems

Lesson Vocabulary

Problem 1 Solving a Matrix Equation Using an Inverse Matrix

Think

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