Let $M_{ij}$ and $A_{ij}$ denote respectively minors and co-factors of the element in the $i$th row and $j$th column of the matrix $\begin{bmatrix}1&2&-1\\3&2&3\\4&-1&0\end{bmatrix}$. Then

(A) $M_{32} = 6$
(B) $M_{23} = 9$
(C) $A_{32}=-6$
(D) $A_{23}=-9$

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (1) → (A) and (C) only

Matrix:

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

Minor $M_{32}$ (3rd row, 2nd column): delete row 3 and column 2

$M_{32}=\begin{vmatrix}1 & -1 \\ 3 & 3\end{vmatrix} =1\cdot 3 - (-1)\cdot 3 = 3+3 = 6$

Minor $M_{23}$ (2nd row, 3rd column): delete row 2 and column 3

$M_{23}=\begin{vmatrix}1 & 2 \\ 4 & -1\end{vmatrix} =1\cdot (-1) - 2\cdot 4 = -1 - 8 = -9$

Cofactor $A_{32}$:

$A_{32}=(-1)^{3+2}M_{32}=(-1)^{5}\cdot 6 = -6$

Cofactor $A_{23}$:

$A_{23}=(-1)^{2+3}M_{23}=(-1)^{5}\cdot (-9)=9$

Correct statements: (A), (C).