Question

The cofactor of the element \( a_{21} \) in the expansion of \[ \Delta = \begin{vmatrix} 1 & 4 & 4 -3 & 5 & 9 2 & 1 & 2 \end{vmatrix} \] is

• A. 5
• B. -24
• C. -4
• D. -5

Hint:

The cofactor of an element in a matrix is calculated by multiplying the determinant of its minor by \( (-1)^{i+j} \), where \( i \) and \( j \) are the row and column indices of the element.

Answer 1

The Correct Option is C

Step 1: Understanding the cofactor.
The cofactor of an element \( a_{ij} \) in a matrix is given by: \[ C_{ij} = (-1)^{i+j} \times \text{determinant of the minor of } a_{ij} \]
where the minor of \( a_{ij} \) is the determinant of the submatrix obtained by deleting the \( i \)-th row and the \( j \)-th column from the original matrix.
We are asked to find the cofactor of the element \( a_{21} \), which is the element in the second row and first column of the matrix.

Step 2: Remove the second row and first column to get the minor.
To find the minor of \( a_{21} \), we remove the second row and first column of the matrix: \[ \text{Minor of } a_{21} = \begin{vmatrix} 4 & 4 1 & 2 \end{vmatrix} \]

Step 3: Calculate the determinant of the minor.
Now calculate the determinant of the 2x2 minor matrix:
\[ \text{determinant of minor} = (4 \times 2) - (4 \times 1) = 8 - 4 = 4 \]

Step 4: Calculate the cofactor.
The cofactor is:
\[ C_{21} = (-1)^{2+1} \times 4 = (-1)^3 \times 4 = -4 \]

Step 5: Conclusion.
Thus, the cofactor of \( a_{21} \) is \( -4 \), and the correct answer is option (C).