   Chapter 11.CR, Problem 57E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# 5 3 − 6 0 . Determinants and Inverse Matrices: Find the determinant and, if possible, the inverse of the matrix. [ 3 0 1 2 − 3 0 4 − 2 1 ]

To determine

To find:

The determinant of the matrix , and the inverse, if it exists.

Explanation

Approach:

If A is a square matrix and if the matrix B is obtained from A by adding a multiple of one row to another or a multiple of one column to another, then det(A)=det(B).

If we take a 3×6 matrix with left half as matrix B and right half as I3 and transform the left half into identity matrix by performing elementary row operations on the entire matrix, the right half becomes inverse of the matrix B.

Calculation:

Consider the determinant A=|301230421|.

Perform the transformation R1R132R2.

|301230432011||301230120|

Expand the determinant by third column.

det(A)=1(1)3+3|2312|=(2(2)1(3))=(4+3)=1

Since, det(A)0, the inverse of the matrix exists.

Inverse of the matrix is shown below:

Consider a matrix, [301230421|100010001]

Perform the transformation, R113R1.

[133130131230421|131130130010001][1013230421|1300010001]

Perform the transformation, R2R22R1.

[101322(1)32(0)02(13)421|130002(13)12(0)02(0)001][10130323421|13002310001]

Perform the transformation, R3R34R1.

[1013032344(1)24(0)14(13)|1300231004(13)04(0)14(0)][101303230213|130023104301]

Perform the transformation, R213R2.

[10130(13)3(13)23(13)0213|130023(13)1(13)0(13)4301][101301290213|1300291304301]

Perform the transformation, R3R3+2R2

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