Name: Finding Inverse by Elementary Transformations
Uploaded: 2019-07-19
Duration: 12 min 5 s
Description: In this video, we will learn to find the inverse of matrices using elementary transformations of matrices. We will learn how to find the inverse of a matrix,...

Question

Compute the adjoint of the following matrix:

[\begin{matrix} 2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1 \end{matrix}]

Verify that

(adj A) A = | A | I = A (adj A)

for the above matrix.

Accepted Answer

Given matrix is $C = [\begin{matrix} 2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1 \end{matrix}]$
Now,
$C_{11} = |\begin{matrix} 2 & 5 \\ 4 & - 1 \end{matrix}| = - 22, C_{12} = - |\begin{matrix} 4 & 5 \\ 0 & - 1 \end{matrix}| = 4$ and $C_{13} = |\begin{matrix} 4 & 2 \\ 0 & 4 \end{matrix}| = 16$
$C_{21} = - |\begin{matrix} - 1 & 3 \\ 4 & - 1 \end{matrix}| = 11, C_{22} = |\begin{matrix} 2 & 3 \\ 0 & - 1 \end{matrix}| = - 2$ and $C_{23} = - |\begin{matrix} 2 & - 1 \\ 0 & 4 \end{matrix}| = - 8$
$C_{31} = |\begin{matrix} - 1 & 3 \\ 2 & 5 \end{matrix}| = - 11, C_{32} = - |\begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix}| = 2$ and $C_{33} = |\begin{matrix} 2 & - 1 \\ 4 & 2 \end{matrix}| = 8$
$∴ a d j C = {[\begin{matrix} - 22 & 4 & 16 \\ 11 & - 2 & - 8 \\ - 11 & 2 & 8 \end{matrix}]}^{T} = [\begin{matrix} - 22 & 11 & - 11 \\ 4 & - 2 & 2 \\ 16 & - 8 & 8 \end{matrix}]$

Consider, $(a d j C) C = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
$|C| = 0$
$∴ |C| I = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
and $C (a d j C) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
Thus, $(adj A) A = | A | I = A (adj A)$

Compute the adjoint of the following matrix: $[\begin{matrix} 2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1 \end{matrix}]$
Verify that $(adj A) A = | A | I = A (adj A)$ for the above matrix.

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Compute the adjoint of the following matrix: 2-1342504-1 Verify that adjAA= |A|I=AadjA for the above matrix.

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Compute the adjoint of the following matrix: $[\begin{matrix} 2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1 \end{matrix}]$
Verify that $(adj A) A = | A | I = A (adj A)$ for the above matrix.