Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE?

(A) Minor of $a_{23} = 16$

(B) Minor of $a_{23} = -8$

(C) Cofactor of $a_{23} = -16$

(D) Cofactor of $a_{23} = 8$

(E) Cofactor of $a_{13} = 4$

Choose the correct answer from the options given below:

Given matrix $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$

The element $a_{23}$ (row 2, column 3) = 5

The element $a_{13}$ (row 1, column 3) = 4

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To find the minor of $a_{23}$, cross out row 2 and column 3:

$M_{23} = \begin{vmatrix} 2 & -3 \\ -4 & 2 \end{vmatrix}$

$M_{23} = (2)(2) - (-3)(-4)$

$M_{23} = 4 - 12$

$M_{23} = -8$

Option (B) is TRUE: Minor of $a_{23} = -8$

Option (A) is FALSE

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To find the cofactor of $a_{23}$:

$C_{23} = (-1)^{2+3} \times M_{23}$

$C_{23} = (-1)^{5} \times (-8)$

$C_{23} = (-1) \times (-8)$

$C_{23} = 8$

Option (D) is TRUE: Cofactor of $a_{23} = 8$

Option (C) is FALSE

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To find the cofactor of $a_{13}$, first find the minor by crossing out row 1 and column 3:

$M_{13} = \begin{vmatrix} 0 & 1 \\ -4 & 2 \end{vmatrix}$

$M_{13} = (0)(2) - (1)(-4)$

$M_{13} = 0 + 4$

$M_{13} = 4$

Now find the cofactor:

$C_{13} = (-1)^{1+3} \times M_{13}$

$C_{13} = (-1)^{4} \times 4$

$C_{13} = 1 \times 4$

$C_{13} = 4$

Option (E) is TRUE: Cofactor of $a_{13} = 4$

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Therefore, statements (B), (D), and (E) are TRUE.