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Q.
$A$ be a square matrix of order 2 with $|A| \neq 0$ such that $|A+| A|$ adj $(A)|=0,$ where adj $(A)$ is a adjoint of matrix $A$ then the value of $\ A-|A| $ adj $(A) \mid$ is
Matrices
Solution:
Let $A=\begin{bmatrix}m&n\\ p&q\end{bmatrix}$, adj $(a)=\begin{bmatrix}p&-n\\ -p&m\end{bmatrix}$
Let $|A|=d=m q-n p$
$|A+d \operatorname{adj} A|= \begin{vmatrix}m+qd&n\left(1-d\right)\\ p\left(1-d\right)&q+md\end{vmatrix}=0$
$\Rightarrow m q+m^{2} d+q^{2} d+m q d^{2}-n p+2 n p d-n p d^{2}=0$
$\Rightarrow (m q-n p)+(m q-n p) d^{2}+m^{2} d+q^{2} d+2 m q d-2 d^{2}=0$
$\Rightarrow \left(d+d^{3}-2 d^{2}\right)+d\left(m^{2}+q^{2}+2 m q\right)=0$
$\Rightarrow d\left[(d-1)^{2}+(m+q)^{2}\right]=0 \Rightarrow d=1, m+q=0$
Now, $\mid A-d$ adj $|A|=-(m+q)^{2}+4(m q-n p)=4 d=4$