Which of the following statements are correct?
A. If \( P \) and \( Q \) are square matrices of same order, then \( (PQ^T-QP^T) \) is a skew symmetric matrix.
B. The number of all possible matrices of order \( 3\times 2 \) with each entry either −1 or 1, is 64.
C. If \( P \) is an invertible matrix, then \( P^T \) is also invertible.
D. If \( P \) and \( Q \) are square matrices of the same order, then \( |PQ|=|P|+|Q| \) .
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, B and C only
Using \( (XY)^T=Y^TX^T \) and \( (Q^T)^T=Q \):
\( \displaystyle (PQ^T-QP^T)^T=QP^T-PQ^T \)
\( \displaystyle =-(PQ^T-QP^T) \)
A 3 × 2 matrix has 6 entries, each with 2 choices, so by the multiplication principle the number of matrices is
\( \displaystyle 2^6=64 \)
Also \( |P^T|=|P| \), and the determinant of a product is the product of the determinants, \( |PQ|=|P|\,|Q| \).
So \( PQ^T-QP^T \) changes sign on transposing, there are 64 such matrices, \( |P^T|=|P| \) and \( |PQ|=|P|\,|Q| \).
Checking the statements
(A) If \( P \) and \( Q \) are square matrices of same order, then \( (PQ^T-QP^T) \) is a skew symmetric matrix. — Correct. Its transpose is \( -(PQ^T-QP^T) \).
(B) The number of all possible matrices of order \( 3\times 2 \) with each entry either −1 or 1, is 64. — Correct. \( 2^6=64 \).
(C) If \( P \) is an invertible matrix, then \( P^T \) is also invertible. — Correct. \( |P^T|=|P|\ne 0 \).
(D) If \( P \) and \( Q \) are square matrices of the same order, then \( |PQ|=|P|+|Q| \) . — Incorrect. \( |PQ|=|P|\,|Q| \); for P = Q = I of order 2, |PQ| = 1 but |P| + |Q| = 2.