Let P and Q be 3 x 3 matrices with P \neq 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

 

Answers (2)

As we learnt in 

Property of matrix multiplication -

It is associative .

- wherein

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

Given 

\Rightarrow P^{3}=Q^{3} \Rightarrow P^{2}Q=Q^{2}P \Rightarrow PQ^{2}=P^{2}Q

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

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

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

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 neha

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