$A$ is a square matrix of order 2 with $|A|\not =0$ such that $|A+|A|\text {adj} (A)|=0$, then find $|A-|A|\text {adj} (A)|$

Question

If $A=\begin{bmatrix} m&n \\p&q \end{bmatrix}$

Then the expression become $$\begin{vmatrix} m+ad& n(1-d) \\ p(1-d)&q+md \end{vmatrix}

Where $d=|A|=mq-np$

So

$$(mq-np) +d^2(mq-np) +m^2d +q^2d+2npd=0$$

I tried simplifying it but I don’t find the results useful. Is this the right way to do this? Is there a way where I don’t have to take $m,n,p,q$ as variables just use the properties of matrices and determinants?

Answer 1

Let $I$ be the identity. From $A\operatorname{adj}(A)=|A|I=dI$ we know -- as $d\neq0$ -- that $d\cdot\operatorname{adj}(A)=d^2A^{-1}$. Hence $$ \begin{align} A-d\operatorname{adj}(A)&=A-d^2A^{-1}\\ &=A^{-1}(A^2-d^2I)\\ &=A^{-1}(A+dI)(A-dI). \end{align} $$