Question

If \( b \) and \( c \) are non-zero real numbers, and \[ A = \begin{bmatrix} 1 & b & c \\ b & 2 & 3 \\ c & 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & b & c \\ -b & 0 & 2 \\ -c & -2 & 0 \end{bmatrix}, \] then \( \det(A + B) = \) ?

• A. 3
• B. 1
• C. -1
• D. 0

Hint:

For determinant of sum \( A + B \), first simplify the matrix and then use standard determinant methods such as cofactor expansion.

Answer 1

The Correct Option is A

We are to compute: \[ A + B = \begin{bmatrix} 1+0 & b+b & c+c\\ b - b & 2 + 0 & 3 + 2 \\ c - c & 3 - 2 & 4 + 0 \end{bmatrix} = \begin{bmatrix} 1 & 2b & 2c \\ 0 & 2 & 5 \\ 0 & 1 & 4 \end{bmatrix} \] Now, compute the determinant: \[ \det(A + B) = \begin{vmatrix} 1 & 2b & 2c \\ 0 & 2 & 5 \\ 0 & 1 & 4 \end{vmatrix} \] Use cofactor expansion along the first column: \[ = 1 \cdot \begin{vmatrix} 2 & 5 \\ 1 & 4 \end{vmatrix} = 1(2 \cdot 4 - 5 \cdot 1) = 1(8 - 5) = 3 \]