# Let P and Q be 3 x 3 matrices with P Q. If P3 = Q3 and P2Q = Q2P , then determinant of (P2 + Q2) is equal to Option 1) -2 Option 2) 1 Option 3) 0 Option 4) -1

As we learnt in

Property of matrix multiplication -

It is associative .

- wherein

$\left ( AB \right )C=A\left ( BC \right )$

Given

$\dpi{100} \Rightarrow P^{3}=Q^{3} \Rightarrow P^{2}Q=Q^{2}P \Rightarrow PQ^{2}=P^{2}Q$

$\dpi{100} \Rightarrow \; \; P^{3}+PQ^{2}=Q^{3}+P^{2}Q$

$\dpi{100} \Rightarrow \; \; P(P^{2}+Q^{2})=(P^{2}+Q^{2})Q$

$\dpi{100} \Rightarrow P\neq Q\ so that\ \Rightarrow P^{2}+Q^{2}\; is \; singular.$

$\dpi{100} Hence,\; \; \left | P^{2}+Q^{2} \right |=0$

Option 1)

-2

This option is incorrect.

Option 2)

1

This option is incorrect.

Option 3)

0

This option is correct.

Option 4)

-1

This option is incorrect.

N

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