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:

(B), (D) and (E) only

The correct answer is Option (3) → (B), (D) and (E) only

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

Minor of $a_{23}$

Delete row $2$ and column $3$

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

$=4-12=-8$

Hence

(B) is true, (A) is false

Cofactor of $a_{23}$

$A_{23}=(-1)^{2+3}M_{23}=-(-8)=8$

Hence

(D) is true, (C) is false

Cofactor of $a_{13}$

Delete row $1$ and column $3$

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

$A_{13}=(-1)^{1+3}M_{13}=4$

Hence

(E) is true

The correct options are (B), (D) and (E).