Let A and B be matrices such that the sums and products below are defined, and let k be a scalar. Consider the following statements:
(A) \((A')' = A\)
(B) \((kA)' = kA'\)
(C) \((A + B)' = A' + B'\)
(D) \((AB)' = A'B'\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
The transpose A' is obtained by turning the rows of A into columns. Taking the transpose of a sum or of a scalar multiple works element by element, but for a product the order is reversed.
So \((A')' = A\), \((kA)' = kA'\), \((A + B)' = A' + B'\) and \((AB)' = B'A'\).
Checking the statements
(A) \((A')' = A\) — Correct. Turning rows into columns twice gives back A.
(B) \((kA)' = kA'\) — Correct. Every element is multiplied by k before or after the transpose.
(C) \((A + B)' = A' + B'\) — Correct. Sums are taken element by element.
(D) \((AB)' = A'B'\) — Incorrect. The rule is \((AB)' = B'A'\); for A of order \(2 \times 3\) and B of order \(3 \times 1\), \(A'B'\) is not even defined.