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

Step 1: Concept
The minor $M_{ij}$ of an element $a_{ij}$ in a matrix is the determinant of the submatrix formed by deleting the $i^{th}$ row and $j^{th}$ column.

Step 2: Meaning
For the element $a_{23}$, we need to remove the second row and the third column of matrix $A$ and find the determinant of the remaining $2 \times 2$ matrix.

Step 3: Analysis
Removing Row 2 ([4, 5, 6]) and Column 3 ([3, 6, 9]) leaves us with: $M_{23} = \begin{vmatrix} 1 & 2 \\7 & 4 \end{vmatrix}$. Calculating the determinant: $(1 \times 4) - (2 \times 7) = 4 - 14 = -10$.

Step 4: Conclusion
The value of the minor $M_{23}$ is -10.
Final Answer: (B)