If \(A =\begin{bmatrix}5&6\\3&2\end{bmatrix}\) then which of the following is correct?
(A) \(|A|\) is positive
(B) \(|\mathrm{adj}\, A| = -8\)
(C) Cofactor of 3 is 6
(D) \(|2A| = -32\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (D) only
\( |A| = 5 \times 2 - 6 \times 3 = -8 \)
For a 2 × 2 matrix, \(|\mathrm{adj}\,A| = |A|^{2-1} = |A|\) and \(|kA| = k^2|A|\):
\( |\mathrm{adj}\,A| = -8 \)
\( |2A| = 2^2 \times (-8) = -32 \)
The element 3 is in row 2, column 1, so its cofactor is its minor 6 with the sign \((-1)^{2+1}\):
\( A_{21} = (-1)^{2+1} \times 6 = -6 \)
So \(|A| = -8\), \(|\mathrm{adj}\,A| = -8\), the cofactor of 3 is \(-6\) and \(|2A| = -32\).
Checking the statements
(A) \(|A|\) is positive — Incorrect. \(|A| = -8\).
(B) \(|\mathrm{adj}\, A| = -8\) — Correct. \(|\mathrm{adj}\,A| = |A| = -8\).
(C) Cofactor of 3 is 6 — Incorrect. 6 is the minor; the cofactor is \(-6\).
(D) \(|2A| = -32\) — Correct. \(4 \times (-8) = -32\).