If A and B are square matrices of same order n, then identify correct statements from the statements given below:
A. \(|\mathrm{adj}\,A| = |A|^{n-1}\)
B. \(|A \cdot B| = |B| \cdot |A|\)
C. \(\mathrm{adj}\,A' = (\mathrm{adj}\,A)'\)
D. \(\mathrm{adj}\,AB = (\mathrm{adj}\,A) \cdot (\mathrm{adj}\,B)\)
E. \(|A^n| = |A|^n\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, B, C and E only
For square matrices A, B of order n, the standard properties are
\(|AB| = |A|\,|B|\)
\(|\mathrm{adj}\,A| = |A|^{n-1}\)
\(\mathrm{adj}(AB) = (\mathrm{adj}\,B)(\mathrm{adj}\,A)\)
So the determinant of a product is the product of determinants, but the adjoint of a product reverses the order.
Checking the statements
(A) \(|\mathrm{adj}\,A| = |A|^{n-1}\) — Correct. This is the standard result for a matrix of order n.
(B) \(|A \cdot B| = |B| \cdot |A|\) — Correct. \(|AB| = |A|\,|B|\), and numbers commute.
(C) \(\mathrm{adj}\,A' = (\mathrm{adj}\,A)'\) — Correct. The cofactors of A’ are the cofactors of A with rows and columns swapped.
(D) \(\mathrm{adj}\,AB = (\mathrm{adj}\,A) \cdot (\mathrm{adj}\,B)\) — Incorrect. The order reverses: \(\mathrm{adj}(AB) = (\mathrm{adj}\,B)(\mathrm{adj}\,A)\).
(E) \(|A^n| = |A|^n\) — Correct. Using \(|AB| = |A|\,|B|\) repeatedly on \(A \cdot A \cdots A\).