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  4. If A and P are different matrices of order n satisfying A3=P3 and A2 P=P2 A (where .|A-P| ≠ 0) then mid A2+P2 is equal to

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Q. If $A$ and $P$ are different matrices of order $n$ satisfying $A^{3}=P^{3}$ and $A^{2} P=P^{2} A$ (where $\left.|A-P| \neq 0\right)$ then $\mid A^{2}+P^{2}$ is equal to

Matrices

Solution:

$\left(A^{2}+P^{2}\right)(A-P) =A^{3}-A^{2} P+P^{2} A-P^{3} $
$=\left(A^{3}-P^{3}\right)+\left(P^{2} A-A^{2} P\right) $
$=0 $
$\therefore \,\,\,\mid\left(A^{2}+P^{2}\right)(A-P) \mid=0 $
$\therefore \,\,\, \left|A^{2}+P^{2}\right|=0 \,\, (\because|A-P| \neq 0) $