If \(A=\begin{bmatrix}5&2\\4&3\end{bmatrix}\) is a given matrix, then which of the following statements are correct?
(A) \(|A|=7\)
(B) minor of \(3=-5\)
(C) co-factor of \(2=-4\)
(D) \(\mathrm{adj}(A) =\begin{bmatrix}3&-2\\-4&5\end{bmatrix}\)
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
The determinant:
\(|A| = 5 \times 3 - 2 \times 4 = 15 - 8 = 7\)
The minor of an entry is the determinant left after deleting its row and column; the cofactor is the minor times \((-1)^{i+j}\).
Entry 3 (row 2, column 2): minor \(= 5\)
Entry 2 (row 1, column 2): minor \(= 4\), cofactor \(= (-1)^{1+2} \times 4 = -4\)
The cofactors of 5, 2, 4, 3 are 3, −4, −2, 5. The adjoint is the transpose of the cofactor matrix:
\( \displaystyle \mathrm{adj}(A) = \begin{bmatrix}3&-4\\-2&5\end{bmatrix}^{T} = \begin{bmatrix}3&-2\\-4&5\end{bmatrix} \)
So \(|A| = 7\), the minor of 3 is 5, the cofactor of 2 is −4, and \(\mathrm{adj}(A)\) has rows (3, −2) and (−4, 5).
Checking the statements
(A) \(|A|=7\) — Correct. \(15 - 8 = 7\).
(B) minor of \(3=-5\) — Incorrect. The minor of 3 is 5.
(C) co-factor of \(2=-4\) — Correct. \((-1)^{1+2} \times 4 = -4\).
(D) \(\mathrm{adj}(A) =\begin{bmatrix}3&-2\\-4&5\end{bmatrix}\) — Correct. Swap the diagonal entries and change the signs of the other two.