Question

If \[ B = \begin{bmatrix} 3 & \alpha & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \] is the adjoint of a 3x3 matrix \( A \) and \( |A| = 4 \), then \( \alpha \) is equal to:

• A. 1
• B. 0
• C. -1
• D. -2

Hint:

The adjoint matrix \( B \) is the transpose of the cofactor matrix. The elements of \( B \) are cofactors corresponding to the elements of \( A \).

Answer 1

The Correct Option is A

We are given that \( B \) is the adjoint of a matrix \( A \) and the determinant of \( A \), \( |A| = 4 \). The adjoint \( B \) is related to the determinant of \( A \) as follows: \[ B = \text{adj}(A) = |A| \times A^{-1} \] The elements of \( B \) are cofactors of the corresponding elements in \( A \), so the element \( B_{12} \) (second element in the first row) is the cofactor of \( A_{12} \), which is \( \alpha \). The cofactor \( \alpha \) is equal to \( |A| \) multiplied by the corresponding minor of \( A_{12} \). From the question, we are told that \( |A| = 4 \). For \( \alpha \), we can compute the cofactor corresponding to the matrix element. The matrix \( B \) suggests that for \( A \), the cofactor corresponding to \( A_{12} \) (which is \( \alpha \)) is 1.
Conclusion: Therefore, \( \alpha = 1 \).