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Properties of Determinants

Let P and Q b...

Question

Let P and Q be $3 α 3$ matrices with $P \neq Q$ . If $P^{3} = Q^{3}$ and $P^{2} Q = Q^{2} P$ , then | $P^{2} + Q^{2} | =$

A

- 2

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B

1

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C

0

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D

- 1

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Solution

The correct option is B $0$
From given, we have,

$P^{3} = Q^{3}$

$P^{3} - P^{2} Q = Q^{3} - Q^{2} P$

$P^{2} (P - Q) + Q^{2} (P - Q) = 0$

$(P - Q) (P^{2}) + (P - Q) (Q^{2}) = 0$

$(P^{2} + Q^{2}) (P - Q) = 0$

$\Rightarrow | P^{2} + Q^{2} | = 0$

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Similar questions

Q. Let P and Q be

3 \times 3

matrices such that

P \neq Q .

P^{3} = Q^{3} a n d P^{2} Q = Q^{2} P,

then determinant of

(P^{2} + Q^{2})

P

Q

3 \times 3

P \neq Q

P^{3} = Q^{3}

P^{2} Q = Q^{2} P

, then determinant of

(P^{2} + Q^{2})

Q. Let P and Q be

3 \times 3

matrices such that

P \neq Q .

P^{3} = Q^{3} a n d P^{2} Q = Q^{2} P,

then determinant of

(P^{2} + Q^{2})

Q. Let P and Q be

3 \times 3

matrices such that

P \neq Q

P^{3} = Q^{3}

P^{2} Q = Q^{2} P

then determinant of

(P^{2} + Q^{2})

P

Q

be 2*2 matrices. Consider the statements.

P Q = 0 \Rightarrow 1) P = 0

Q = 0

b o t h

P Q = I_{2} \Rightarrow P = Q^{- 1}

(P + Q)^{2} = P^{2} + 2 P Q + Q^{2}

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