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 & 0 & - 1 \\ 5 & 1 & 0 \\ 1 & 1 & 3 \end{matrix}]

Verify that

(a d j A) A = | A | I = A (a d j A)

for the above matrix.

Accepted Answer

Given matrix is $D = [\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 1 & 1 & 3 \end{matrix}]$
Now,
$C_{11} = |\begin{matrix} 1 & 0 \\ 1 & 3 \end{matrix}| = 3, C_{12} = - |\begin{matrix} 5 & 0 \\ 1 & 3 \end{matrix}| = - 15$ and $C_{13} = |\begin{matrix} 5 & 1 \\ 1 & 1 \end{matrix}| = 4$
$C_{21} = - |\begin{matrix} 0 & - 1 \\ 1 & 3 \end{matrix}| = - 1, C_{22} = |\begin{matrix} 2 & - 1 \\ 1 & 3 \end{matrix}| = 7$ and $C_{23} = - |\begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix}| = - 2$
$(a d j A) A = | A | I = A (a d j A)$ and $C_{33} = |\begin{matrix} 2 & 0 \\ 5 & 1 \end{matrix}| = 2$
$∴ a d j D = {[\begin{matrix} 3 & - 15 & 4 \\ - 1 & 7 & - 2 \\ 1 & - 5 & 2 \end{matrix}]}^{T} = [\begin{matrix} 3 & - 1 & 1 \\ - 15 & 7 & - 5 \\ 4 & - 2 & 2 \end{matrix}]$

Consider, $(a d j D) D = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}]$
and $|D| = 2$
$∴ |D| I = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}]$
and $D (a d j D) = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}]$
Thus, $(a d j A) A = | A | I = A (a d j A)$

Compute the adjoint of the following matrix: $[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 1 & 1 & 3 \end{matrix}]$
Verify that $(a d j A) A = | A | I = A (a d j A)$ 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:20-1510113 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: $[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 1 & 1 & 3 \end{matrix}]$
Verify that $(a d j A) A = | A | I = A (a d j A)$ for the above matrix.