Let $A = \begin{bmatrix}2&-1&3\\1&2&-1\\4&1&2\end{bmatrix}$. $M_{ij}$ and $A_{ij}$ respectively denote the minor and cofactor of an element $a_{ij}$ of matrix $A = [a_{ij}]$.

(A) $M_{23} = 6$
(B) $A_{22}=-8$
(C) $A_{13} = 7$
(D) $M_{32} = -5$

Choose the correct answer from the options given below:

(A), (B) and (D) only

The correct answer is Option (4) → (A), (B) and (D) only

(A) $M_{23} = 6$
(B) $A_{22}=-8$
(D) $M_{32} = -5$

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

(A) $M_{23}$: Minor of element in 2nd row, 3rd column

Delete 2nd row and 3rd column: $\begin{vmatrix} 2 & -1 \\ 4 & 1 \end{vmatrix} = 2(1) - (-1)(4) = 2 + 4 = 6$

(B) $A_{22}$: Cofactor of element in 2nd row, 2nd column

Delete 2nd row and 2nd column: $\begin{vmatrix} 2 & 3 \\ 4 & 2 \end{vmatrix} = 2(2) - 3(4) = 4 - 12 = -8$ Cofactor sign for (2,2) is $(-1)^{2+2} = +1$, so $A_{22} = -8$

(C) $A_{13}$: Cofactor of element in 1st row, 3rd column

Delete 1st row and 3rd column: $\begin{vmatrix} 1 & 2 \\ 4 & 1 \end{vmatrix} = 1(1) - 2(4) = 1 - 8 = -7$ Cofactor sign for (1,3) is $(-1)^{1+3} = +1$, so $A_{13} = -7$

(D) $M_{32}$: Minor of element in 3rd row, 2nd column

Delete 3rd row and 2nd column: $\begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = 2(-1) - 3(1) = -2 - 3 = -5$

Correct statements: (A), (B), (D)