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) $adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$

Choose the correct answer from the options given below:

Given matrix $A = \begin{bmatrix} 5 & 2 \\ 4 & 3 \end{bmatrix}$

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Checking Statement (A): $|A| = 7$

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

$|A| = (5)(3) - (2)(4)$

$|A| = 15 - 8$

$|A| = 7$

Statement (A) is correct.

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Checking Statement (B): minor of $3 = -5$

Element 3 is in row 2, column 2.

Deleting row 2 and column 2 leaves only element 5.

Minor of $3 = 5$

Statement (B) is incorrect.

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Checking Statement (C): co-factor of $2 = -4$

Element 2 is in row 1, column 2 (so $i=1$, $j=2$).

Deleting row 1 and column 2 leaves only element 4.

Minor of $2 = 4$

Cofactor $= (-1)^{i+j} \times \text{minor}$

Cofactor $= (-1)^{1+2} \times 4$

Cofactor $= (-1)^3 \times 4$

Cofactor $= -4$

Statement (C) is correct.

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Checking Statement (D): $adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$

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

$adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$

Statement (D) is correct.

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Statements (A), (C), and (D) are correct.