Question

To 0 11
Show that matrix [0 01][010][100]
is invertible. Find its inverse by adjoint
method.
​

Answer 1

Inverse of matrix is $\left[\begin{array}{ccc}0&0& 1\\0& 1&0\\ 1&0&0\end{array}\right]$

Given:

• A 3×3 matrix.
• $\left[\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right]$

To find:

• Show that matrix is invertible.
• Find its inverse by Adjoint method.

Solution:

Concept/Formula to be used:

• Non-Singular matrix can be invertible. A Non-singular matrix's determinant is a non zero quantity.
• $A^{-1}= \frac{Adj.( A )}{ |A| }$

Step 1:

Find determinant of matrix.

Let the matrix is A.

$|A|=\left |\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right | = 0(0 - 0) - 0(0 - 0) + 1(0 - 1)$

or

$|A| = - 1$

As,

$\bf |A| \neq0$

Thus,

Matrix is non-singular and it is invertible.

Step 2:

To find inverse of matrix.

First find Minor matrix.

Minor element for every element can be find by hiding that particular row and column and find determinant of remaining 2×2 matrix.

$M_{(1,1)}=\left |\begin{array}{cc}1&0\\0&0\end{array}\right | = 0$

by this way find all minor.

$M = \left[\begin{array}{ccc}0&0& - 1\\0& - 1&0\\ - 1&0&0\end{array}\right]$

Step 3:

Find Co-factor matrix (C).

It can be find by multiplying each element of minor matrix by

$(-1)^{i+j}$

So,

$C = \left[\begin{array}{ccc}0&0& - 1\\0& - 1&0\\ - 1&0&0\end{array}\right]$

Step 4:

Find Adjoint matrix.

Adjoint matrix can be find by transpose of Co-factor matrix.

$Adj.(A) = \left[\begin{array}{ccc}0&0& - 1\\0& - 1&0\\ - 1&0&0\end{array}\right]$

Step 5:

Find inverse of matrix.

$A^{-1}= \frac{Adj.( A )}{ |A| }$

or

$A ^{ - 1} = \left[\begin{array}{ccc}0&0& 1\\0& 1&0\\ 1&0&0\end{array}\right]$

Thus,

Inverse of matrix is

$\bf A ^{ - 1} = \left[\begin{array}{ccc}0&0& 1\\0& 1&0\\ 1&0&0\end{array}\right]$

Learn more:

1) find the inverse of the matrix

( 1 2 1 )

( 3 0 1 )

( 0 2 1 )

using adjoint method.

https://brainly.in/question/43031277

2) find the inverse of matrix A by using adjoint method where a is equals to

(1 0 1)

A= (0 2 3)

(1 2 1)

https://brainly.in/question/40469452