Let A and B be square matrices of order 3 and k is a constant. If \(|A| \ne 0\) and \(|B| \ne 0\), where |A| represents the determinant of A, then which of the following statements are true? (where A' denotes the transpose of matrix A)
(A) \((AB)^{-1}=B^{-1}A^{-1}\)
(B) \((A+B)'=A'\times B'\)
(C) \((AB)'=A'B'\)
(D) \(|kA|=k^3|A|\)
(E) \(|A'|=|A|\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (D), (E) only
Since \(|A| \ne 0\) and \(|B| \ne 0\), both A and B are invertible. The standard results for square matrices of order 3 are:
\((AB)^{-1} = B^{-1}A^{-1}\)
\((A+B)' = A' + B'\)
\((AB)' = B'A'\)
\(|kA| = k^3|A|\)
\(|A'| = |A|\)
So the inverse and the transpose of a product reverse the order, the transpose of a sum is the sum of the transposes, and for order 3, \(|kA| = k^3|A|\) and \(|A'| = |A|\).
Checking the statements
(A) \((AB)^{-1}=B^{-1}A^{-1}\) — Correct. The reversal law for inverses.
(B) \((A+B)'=A'\times B'\) — Incorrect. The transpose of a sum is \(A' + B'\), not a product.
(C) \((AB)'=A'B'\) — Incorrect. The order reverses: \((AB)' = B'A'\).
(D) \(|kA|=k^3|A|\) — Correct. Each of the 3 rows of kA has the factor k.
(E) \(|A'|=|A|\) — Correct. Interchanging rows and columns does not change a determinant.