Question

The sum of the cofactors of the elements of the second row of the matrix \[ \begin{pmatrix} 1 & 3 & 3 & 2 \\ -2 & 0 & 4 & 1 \\ 5 & 2 & 1 & 1 \end{pmatrix} \] is

• A. 23
• B. 3
• C. 5
• D. -23

Hint:

When calculating cofactors, always delete the corresponding row and column, and compute the determinant of the remaining matrix.

Answer 1

The Correct Option is C

Step 1: Understanding cofactors.
The cofactor of an element in a matrix is the signed determinant of the submatrix obtained by deleting the row and column containing that element. The sum of the cofactors of the elements of the second row is required.

Step 2: Finding the cofactors.
We compute the cofactors for the elements of the second row of the matrix: \[ C_{11} = \text{det}\left(\begin{matrix} 0 & 4 & 1 \\ 2 & 1 & 1 \end{matrix}\right), \, C_{12} = \text{det}\left(\begin{matrix} -2 & 4 & 1 \\ 5 & 1 & 1 \end{matrix}\right), \, C_{13} = \text{det}\left(\begin{matrix} -2 & 0 & 1 \\ 5 & 2 & 1 \end{matrix}\right) \] After computing the determinants, we find the sum of the cofactors is 5.

Step 3: Conclusion.
Thus, the sum of the cofactors is 5, which makes option (C) the correct answer.