Question

In the matrix $A=\begin{bmatrix}1& 2& 3 \\ 4& 5 & 6 \\ 7& 4 & 9\end{bmatrix}$, the minor $M_{23}$ of the $a_{23}$ is

• A. 10
• B. -10
• C. -6
• D. 6

Hint:

Minor is just the determinant of what's left after crossing out the element's row and column. Don't confuse it with the cofactor, which requires a sign change.

Answer 1

The Correct Option is B

To determine the minor \( M_{23} \) of the element \( a_{23} \) in matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 4 & 9 \end{bmatrix} \), we first need to understand the concept of a "minor" in a matrix.

The minor of an element \( a_{ij} \) in a matrix is the determinant of the submatrix that remains after removing the \( i \)-th row and \( j \)-th column from the matrix.

Here, we are asked to find the minor \( M_{23} \) for the element \( a_{23} \) (which is 6, the element in the second row and third column of the matrix).

1. First, remove the second row and third column from matrix \( A \), resulting in the submatrix: 

1	2
7	4

1. The minor \( M_{23} \) is the determinant of this submatrix: 

\[\begin{vmatrix} 1 & 2 \\ 7 & 4 \end{vmatrix}\]

1. Calculate this determinant using the formula \( ad - bc \) for a \( 2 \times 2 \) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\): 

\[\begin{vmatrix} 1 & 2 \\ 7 & 4 \end{vmatrix} = (1)(4) - (2)(7) = 4 - 14 = -10\]

Therefore, the minor \( M_{23} \) is \( -10 \).