KCET · Maths · Matrices
If \( P \) and \( Q \) are symmetric matrices of the same order then \( P Q-Q P \) is
- A symmetric matrix
- B skew symmetric matrix
- C identity matrix
- D zero matrix
Answer & Solution
Correct Answer
(B) skew symmetric matrix
Step-by-step Solution
Detailed explanation
(B)
Given \(P=P^{\prime}\) and \(Q=Q^{\prime}\)
\((P Q-Q P)^{\prime}=(P Q)^{\prime}-(Q P)^{\prime}\)
\(=\left(Q^{\prime} P^{\prime}-P^{\prime} Q^{\prime}\right)\)
\(=Q P-P Q\)
\(=-[P Q-Q P]\)
\(\therefore(P Q-Q P)\) is skew symmetric.
Given \(P=P^{\prime}\) and \(Q=Q^{\prime}\)
\((P Q-Q P)^{\prime}=(P Q)^{\prime}-(Q P)^{\prime}\)
\(=\left(Q^{\prime} P^{\prime}-P^{\prime} Q^{\prime}\right)\)
\(=Q P-P Q\)
\(=-[P Q-Q P]\)
\(\therefore(P Q-Q P)\) is skew symmetric.
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