Let A and B be square matrices of the same order. Which of the following statements are correct?
(A) \((AB)' = B'A'\)
(B) If A and B are skew-symmetric matrices, then \(3A + B\) is a skew-symmetric matrix.
(C) \(A + A'\) is a skew-symmetric matrix.
(D) \(AA'\) is a skew-symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
The rules of the transpose are
\((A')' = A,\quad (kA)' = kA'\)
\((A + B)' = A' + B',\quad (AB)' = B'A'\)
A matrix M is symmetric if \(M' = M\) and skew-symmetric if \(M' = -M\). Using the rules:
\((3A + B)' = 3A' + B'\)
\((A + A')' = A' + A\)
\((AA')' = (A')'A' = AA'\)
So \((AB)' = B'A'\), \((3A + B)' = 3A' + B'\), and both \(A + A'\) and \(AA'\) are equal to their own transposes.
Checking the statements
(A) \((AB)' = B'A'\) — Correct. This is the reversal rule for the transpose of a product.
(B) If A and B are skew-symmetric matrices, then \(3A + B\) is a skew-symmetric matrix. — Correct. With \(A' = -A\) and \(B' = -B\), \((3A + B)' = -3A - B = -(3A + B)\).
(C) \(A + A'\) is a skew-symmetric matrix. — Incorrect. \((A + A')' = A + A'\), so it is symmetric.
(D) \(AA'\) is a skew-symmetric matrix. — Incorrect. \((AA')' = AA'\), so it is symmetric.