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  4. If A=[a b c d] such that A satisfies the relation A2-(a+d) A=O, then inverse of A is

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Q. If $A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ such that $A$ satisfies the relation $A^2-(a+d) A=O$, then inverse of $A$ is

Matrices

Solution:

Suppose $A^{-1}$ exists, then
$A^{-1} A^2-(a+d) A^{-1} A=O $
$\Rightarrow A-(a+d) I=O $
$\Rightarrow\begin{bmatrix}-d & b \\c & -a\end{bmatrix}=O $
$\Rightarrow a=b=c=d=0 \Rightarrow|A|=0$
$A$ contradiction. Thus, $A^{-1}$ does not exist.