If $A = \begin{bmatrix} 5 & 6 \\ 3 & 2 \end{bmatrix}$ then which of the following is correct?

(A) $|A|$ is positive

(B) $|adj\ A| = -8$

(C) Cofactor of 3 is 6

(D) $|2A| = -32$

Choose the correct answer from the options given below:

Given: $A = \begin{bmatrix} 5 & 6 \\ 3 & 2 \end{bmatrix}$

For a 2×2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$

$|A| = (5)(2) - (6)(3)$

$|A| = 10 - 18$

$|A| = -8$

---

Checking Statement (A): $|A|$ is positive

Since $|A| = -8$ is negative, statement (A) is incorrect.

---

Checking Statement (B): $|adj\ A| = -8$

For any n×n matrix, $|adj(A)| = |A|^{n-1}$

Since A is a 2×2 matrix (n = 2):

$|adj(A)| = |A|^{2-1}$

$|adj(A)| = |A|^1$

$|adj(A)| = -8$

Statement (B) is correct.

---

Checking Statement (C): Cofactor of 3 is 6

The element 3 is at position (row 2, column 1)

The cofactor is $(-1)^{i+j} \times$ Minor

Minor = The element remaining after removing row 2 and column 1 = 6

Sign = $(-1)^{2+1} = (-1)^3 = -1$

Cofactor of 3 = $(-1) \times 6 = -6$

The cofactor is $-6$, not $6$. Statement (C) is incorrect.

---

Checking Statement (D): $|2A| = -32$

For any scalar k and n×n matrix A: $|kA| = k^n|A|$

Since A is 2×2 (n = 2) and k = 2:

$|2A| = 2^2 \times |A|$

$|2A| = 4 \times (-8)$

$|2A| = -32$

Statement (D) is correct.

---

Statements (B) and (D) are correct.