(12) Use the adjoint of matrix to find the third row fourth column element of the inverseof-11432-21411-1Ans. -17/9

Consider the given matrix as A=0-11432-21040110-11. The inverse of the matrix A is given by…

Answered: (12) Use the adjoint of matrix to find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(12) Use the adjoint of matrix to find the third row fourth column element of the inverse
of
-1
1
4
3
2
-2
1
4
1
1
-1
Ans. -17/9

Expert Solution

Step 1

Consider the given matrix as $A = (\begin{matrix} 0 & - 1 & 1 & 4 \\ 3 & 2 & - 2 & 1 \\ 0 & 4 & 0 & 1 \\ 1 & 0 & - 1 & 1 \end{matrix})$ .

The inverse of the matrix A is given by $A^{- 1} = \frac{a d j A}{d e t A}$ .

So, the third row fourth column element of the inverse of A is $\frac{1}{d e t A}$ times the third row fourth column element of the adjoint of A.

Since adjoint of a matrix is the transpose of the cofactor matrix of that matrix, the adjoint of A is the transpose of the cofactor matrix of A.

Hence, the third row fourth column element of the adjoint of A is the fourth row third column element of the cofactor matrix of A.

Step 2

The fourth row third column element of the cofactor matrix of A is given by the product of ${(- 1)}^{4 + 3}$ and the determinant of the matrix obtained from A by removing its fourth row and third column.

The matrix obtained from A by removing its fourth row and third column is $(\begin{matrix} 0 & - 1 & 4 \\ 3 & 2 & 1 \\ 0 & 4 & 1 \end{matrix})$ .

Therefore,

$\begin{array}{rcl} {(- 1)}^{4 + 3} |\begin{matrix} 0 & - 1 & 4 \\ 3 & 2 & 1 \\ 0 & 4 & 1 \end{matrix}| & = & {(- 1)}^{4 + 3} \{0 [(2) (1) - (4) (1)] - (- 1) [(3) (1) - (0) (1)] + 4 [(3) (4) - (0) (2)]\} \\ = & {(- 1)}^{4 + 3} \{0 + 3 + 48\} \\ = & - 1 \times 51 \\ = & - 51 \end{array}$

Hence, the fourth row third column element of the cofactor matrix of A is -51.

That is, the third row fourth column element of the adjoint of A is -51.

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