For the matrix \(A = \begin{bmatrix}2&-1&-1\\0&2&3\\1&-2&1\end{bmatrix}\), which of the following statements are correct?
(A) The order of the matrix is \(3 \times 3\)
(B) \(|A| = 21\)
(C) \(|\mathrm{adj}\,A| = 225\)
(D) A is skew symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), and (C) only
A has 3 rows and 3 columns. Expanding |A| along the first row:
\( |A| = 2\begin{vmatrix}2&3\\-2&1\end{vmatrix} - (-1)\begin{vmatrix}0&3\\1&1\end{vmatrix} + (-1)\begin{vmatrix}0&2\\1&-2\end{vmatrix} \)
\(= 2(2 + 6) + (0 - 3) - (0 - 2)\)
\(= 16 - 3 + 2 = 15\)
For a square matrix of order n, \(|\mathrm{adj}\,A| = |A|^{n-1}\). With n = 3:
\(|\mathrm{adj}\,A| = 15^2 = 225\)
So A is of order \(3 \times 3\), \(|A| = 15\), \(|\mathrm{adj}\,A| = 225\), and the diagonal entries of A are 2, 2, 1.
Checking the statements
(A) The order of the matrix is \(3 \times 3\) — Correct. 3 rows and 3 columns.
(B) \(|A| = 21\) — Incorrect. \(|A| = 15\).
(C) \(|\mathrm{adj}\,A| = 225\) — Correct. \(|A|^{2} = 15^2 = 225\).
(D) A is skew symmetric matrix — Incorrect. A skew-symmetric matrix has all diagonal entries 0, but \(a_{11} = 2\).