Let $A =\begin{bmatrix}1&2&1\\-1&3&2\\2&4&1\end{bmatrix}$ and $M_{ij}, A_{ij}$ respectively denote the minor, co-factor of an element $a_{ij}$ of matrix A, then which of the following are true?

(A) $M_{22} = -1$
(B) $A_{23} = 0$
(C) $A_{32} = 3$
(D) $M_{23} = 1$
(E) $M_{32} = -3$

Choose the correct answer from the options given below:

(A) and (B) only

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

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

(A) $M_{22}$

Delete row $2$, column $2$

$M_{22}=\begin{vmatrix}1&1\\2&1\end{vmatrix}=1-2=-1$

True

(B) $A_{23}$

$M_{23}=\begin{vmatrix}1&2\\2&4\end{vmatrix}=4-4=0$

$A_{23}=(-1)^{2+3}M_{23}=0$

True

(C) $A_{32}$

$M_{32}=\begin{vmatrix}1&1\\-1&2\end{vmatrix}=2+1=3$

$A_{32}=(-1)^{3+2}\cdot3=-3\ne3$

False

(D) $M_{23}=1$

$M_{23}=0$

False

(E) $M_{32}=-3$

$M_{32}=3$

False

The correct statements are (A) and (B).