Let $A=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 2\\ 2& 4& 1\end{array}\right]$ and $M{\mathrm{ij}}_{}$, $A{\mathrm{ij}}_{}$ respectively denote the minor and cofactor of element $a{\mathrm{ij}}_{}$ of matrix A.

Which of the following are true?

(A) $M22=-1$
(B) $A23=0$
(C) $A32=3$
(D) $M23=1$
(E) $M32=-3$

Choose the correct answer:

The correct option is (A) and (B) only.

To determine which of the statements are true, let us first recall the definitions of a minor and a cofactor of an element in a matrix.
Let $a_{ij}$ be the element in the i-th row and j-th column of a square matrix $A$.
1. The minor $M_{ij}$ of the element $a_{ij}$ is the determinant of the submatrix left after deleting the i-th row and j-th column.
2. The cofactor $A_{ij}$ of the element $a_{ij}$ is defined as:
$A_{ij}=(-1{)}^{i+j}{M}_{ij}$

The given matrix $A$ is:
$A=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 2\\ 2& 4& 1\end{array}\right]$

Let us evaluate each of the statements step-by-step:

1. Evaluating Statement (A): $M_{22}=-1$
To find the minor $M_{22}$, we delete the 2nd row and 2nd column of matrix $A$:
$M_{22}=\left|\begin{array}{cc}1& 1\\ 2& 1\end{array}\right|$
Calculating the $2\times 2$ determinant:
$M_{22}=(1\times 1)-(1\times 2)=1-2=-1$
Thus, statement (A) is true.

2. Evaluating Statement (B): $A_{23}=0$
The cofactor $A_{23}$ is given by:
$A_{23}=(-1{)}^{2+3}{M}_{23}=-M_{23}$
First, we find the minor $M_{23}$ by deleting the 2nd row and 3rd column:
$M_{23}=\left|\begin{array}{cc}1& 2\\ 2& 4\end{array}\right|$
Calculating the determinant:
$M_{23}=(1\times 4)-(2\times 2)=4-4=0$
Therefore:
$A_{23}=-0=0$
Thus, statement (B) is true.

3. Evaluating Statement (C) & (E) related to index (3,2):
Let us calculate the minor $M_{32}$ by deleting the 3rd row and 2nd column:
$M_{32}=|\begin{array}{}$ 1$$ 1$\end{array}$$1$$2$|
Calculating the determinant:
$M_{32}=(1\times 2)-(1\times 1)=2-1=1$
This shows that $M_{32}=1$, so statement (E) which states $M_{32}=-3$ is false.
Now let us compute the cofactor $A_{32}$:
$A_{32}=(-1{)}^{3+2}{M}_{32}=-(1)=-1$
Thus, statement (C) which states $A_{32}=3$ is also false.

Conclusion:
Only statements (A) and (B) are correct.