Let A be a square matrix of order 3 with \(|A| = 2\), and let I be the identity matrix of order 3. Which of the following statements are correct?
(A) \(|A^{-1}| = \displaystyle\frac{1}{2}\)
(B) \(|\mathrm{adj}\,A| = 4\)
(C) \(|(A^{T})^{-1}| = 2\)
(D) \(A(\mathrm{adj}\,A) = 2I\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
The basic result is \(A(\mathrm{adj}\,A) = (\mathrm{adj}\,A)A = |A|\,I\). Taking determinants of both sides for order 3:
\(|A|\,|\mathrm{adj}\,A| = |A|^{3}\)
\(|\mathrm{adj}\,A| = |A|^{2}\)
From \(AA^{-1} = I\), \(\displaystyle |A^{-1}| = \frac{1}{|A|}\), and a transpose does not change a determinant.
So with \(|A| = 2\): \(\displaystyle |A^{-1}| = \frac{1}{2}\), \(|\mathrm{adj}\,A| = 4\), \(\displaystyle |(A^{T})^{-1}| = \frac{1}{2}\) and \(A(\mathrm{adj}\,A) = 2I\).
Checking the statements
(A) \(|A^{-1}| = \displaystyle\frac{1}{2}\) — Correct. \(\displaystyle |A^{-1}| = \frac{1}{|A|} = \frac{1}{2}\).
(B) \(|\mathrm{adj}\,A| = 4\) — Correct. \(|A|^{2} = 2^{2} = 4\).
(C) \(|(A^{T})^{-1}| = 2\) — Incorrect. \(\displaystyle |(A^{T})^{-1}| = \frac{1}{|A^{T}|} = \frac{1}{2}\).
(D) \(A(\mathrm{adj}\,A) = 2I\) — Correct. \(A(\mathrm{adj}\,A) = |A|\,I = 2I\).