Explanation Approach: For any square matrix A n , of dimension n × n , then a matrix B n of the same dimension is said to the inverse of A n , only when A n B n = I n also B n A n = I n , here I n is the identity matrix of the same order. Following steps need to be followed in finding the inverse of a matrix: STEP 1: Write the augmented matrix of the form [ A I n ] , where A is the given matrix of dimension n × n and I n is the identity matrix of the same dimension. STEP 2: Apply row transformations so as to transform the obtained augmented matrix in step 1 in the form of [ I n B ] , where ‘ B ’ is the matrix transformed during transformation from the matrix I n taken in the Step 1. STEP 3: Now, the matrix ‘ B ’ is the required inverse matrix of the matrix A . Calculation: Given, [ 1 0 0 2 0 − 2 1 2 2 ] Taking, A = [ 1 0 0 2 0 − 2 1 2 2 ] and I 3 = [ 1 0 0 0 1 0 0 0 1 ] The augmented matrix [ A | I 3 ] , [ 1 0 0 1 0 0 2 0 − 2 0 1 0 1 2 2 0 0 1 ] The first entry of the first column in the above matrix is already 1. Now, let us transform the matrix, so that the remaining entries of the first column become zero by applying row transformations, change the second row such that is added with the first row multiplied by − 2 , [ 1 0 0 1 0 0 2 + ( − 2 ) ( 1 ) 0 + ( − 2 ) ( 0 ) − 2 + ( − 2 ) ( 0 ) 0 + ( − 2 ) ( 1 ) 1 + ( − 2 ) ( 0 ) 0 + ( − 2 ) ( 0 ) 1 2 2 0 0 1 ] R 2 + ( − 2 ) R 1 [ 1 0 0 1 0 0 0 0 − 2 − 2 1 0 1 2 2 0 0 1 ] Chance the third row, such that is added with the first row multiplied by − 1 , [ 1 0 0 1 0 0 0 0 − 2 − 2 1 0 1 + ( − 1 ) ( 1 ) 2 + ( − 1 ) ( 0 ) 2 + ( − 1 ) ( 0 ) 0 + ( − 1 ) ( 1 ) 0 + ( − 1 ) ( 0 ) 1 + ( − 1 ) ( 0 ) ] R 3 + ( − 1 ) R 1 [ 1 0 0 1 0 0 0 0 − 2 − 2 1 0 0 2 2 − 1 0 1 ] The first entry of the second column in the above matrix is already 0

12th Edition

ISBN: 9781305652231

Author: R. David Gustafson, Jeff Hughes

Publisher: Cengage Learning

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Chapter 6.CR, Problem 42E

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The inverse of a matrix [10020−2122].

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Chapter 6.CR, Problem 41E

Chapter 6.CR, Problem 43E

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The inverse of a matrix [ 1 0 0 2 0 − 2 1 2 2 ] .

Solution Summary: The author explains how to find the inverse of a matrix.

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To find:

The inverse of a matrix [10020−2122].

Want to see the full answer?

Chapter 6 Solutions

The inverse of a matrix [ 1 0 0 2 0 − 2 1 2 2 ] .

Solution Summary: The author explains how to find the inverse of a matrix.

Videos

To find: The inverse of a matrix [10020−2122].

Want to see the full answer?

Chapter 6 Solutions

To find:

The inverse of a matrix [10020−2122].