Question

Verify $A\left(adjA\right)=\left(adjA\right)A=|A|I$

$A=\left[\begin{array}{cc}2& 3\\ 4& 6\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}2& 3\\ 4& 6\end{array}\right]$

Here, $|A|=\left|\begin{array}{cc}2& 3\\ 4& 6\end{array}\right|$

$\Rightarrow |A|=12-12=0$

$|A|I=0.I=O$ .....(1)

Now, $adjA=C^T$ where $C$ is the co-factor matrix

$C_{11}=(-1{)}^{1+1}6=6$

$C_{12}=(-1{)}^{1+2}4=-4$

$C_{21}=(-1{)}^{2+1}3=-3$

$C_{22}=(-1{)}^{2+2}2=2$

So, the co-factor matrix C is $\left[\begin{array}{cc}6& -4\\ -3& 2\end{array}\right]$

$\Rightarrow adjA=C^T=\left[\begin{array}{cc}6& -3\\ -4& 2\end{array}\right]$

Now, $A\left(adjA\right)=\left[\begin{array}{cc}2& 3\\ 4& 6\end{array}\right]\left[\begin{array}{cc}6& -3\\ -4& 2\end{array}\right]$

$\Rightarrow A\left(adjA\right)=\left[\begin{array}{cc}12-12& -6+6\\ 24-24& -12+12\end{array}\right]$

$\Rightarrow A\left(adjA\right)=O$ ....(2)

Now, $\left(adjA\right)A=\left[\begin{array}{cc}6& -3\\ -4& 2\end{array}\right]\left[\begin{array}{cc}2& 3\\ 4& 6\end{array}\right]$

$\Rightarrow \left(adjA\right)A=\left[\begin{array}{cc}12-12& 18-18\\ -8+8& -12+12\end{array}\right]$

$\Rightarrow \left(adjA\right)A=O$ .....(3)

From (1), (2) and (3), we have

$A\left(adjA\right)=\left(adjA\right)A=|A|I$

Hence, verified.