Algebra

Matrices & Determinants

Medium

core

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

(A), (B), (C) and (E) only

(A), (D) and (E) only

(A), (C) and (E) only

✅ Correct Option: 1

To find the minor $M_{22}$ , remove the 2nd row and 2nd column from matrix $A$ :

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

This statement is TRUE.

To find the cofactor $A_{23}$ , first find the minor $M_{23}$ by removing the 2nd row and 3rd column:

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

Now, apply the cofactor sign rule $A_{23} = (-1)^{2+3} M_{23}$ :

$A_{23} = (-1) \times 0 = 0$

This statement is TRUE.

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Solution