Question

Find the adjoint and the inverse of the matrix $\left[\begin{array}{ccc}2&1&2\\1&0&1\\2&2&1\end{array}\right]$

Answer 1

Answer:

$adj.A=\left[\begin{array}{ccc}-2&3&1\\1&-2&0\\2&-2&-1\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}-2&3&1\\1&-2&0\\2&-2&-1\end{array}\right]$

Step-by-step explanation:

As we know that Adjoint of matrix is calculated as Minor × Co-factor of each element and taking transpose of it.

or

$adj.A=[A_{ji}]_{n\times n}$

$A=\left[\begin{array}{ccc}2&1&2\\1&0&1\\2&2&1\end{array}\right]\\\\adj.A=\left[\begin{array}{ccc}-2&1&2\\3&-2&-2\\1&0&-1\end{array}\right] ^{'}\\\\adj.A=\left[\begin{array}{ccc}-2&3&1\\1&-2&0\\2&-2&-1\end{array}\right]$

Now

$A^{-1} =\frac{adj.A}{|A|} \\\\|A|= \left|\begin{array}{ccc}2&1&2\\1&0&1\\2&2&1\end{array}\right|\\\\|A|=2(0-2)-1(1-2)+2(2)=1 \\\\\\A^{-1} =\frac{1}{1}\left[\begin{array}{ccc}-2&3&1\\1&-2&0\\2&-2&-1\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}-2&3&1\\1&-2&0\\2&-2&-1\end{array}\right]$