Let \(A = [a_{ij}]\) be a square matrix, where \(a_{ij} = \begin{cases}0, & \text{when } i = j\\1, & \text{otherwise}\end{cases}\). If \(|\mathrm{adj}\,A| = |A|^2\), then which of the following statements are correct?
(A) A is a skew symmetric matrix.
(B) A is a non-singular matrix.
(C) A is a square matrix of order 4.
(D) A is a symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
For a square matrix of order n, \(|\mathrm{adj}\,A| = |A|^{n-1}\). So the condition \(|\mathrm{adj}\,A| = |A|^2\) points to \(n - 1 = 2\), that is, order 3:
\( A = \begin{bmatrix}0&1&1\\1&0&1\\1&1&0\end{bmatrix} \)
\( \displaystyle |A| = 0(0 - 1) - 1(0 - 1) + 1(1 - 0) \)
\( \displaystyle = 0 + 1 + 1 = 2 \)
So A is the symmetric matrix of order 3 above, with \(|A| = 2\).
Checking the statements
(A) A is a skew symmetric matrix. — Incorrect. \(a_{12} = a_{21} = 1\), not \(a_{21} = -a_{12}\).
(B) A is a non-singular matrix. — Correct. \(|A| = 2 \ne 0\).
(C) A is a square matrix of order 4. — Incorrect. \(n - 1 = 2\) gives order 3.
(D) A is a symmetric matrix. — Correct. \(a_{ji} = a_{ij}\) for all i, j.