Question

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

Hint:

Remember the adjoint matrix properties: adjoint \( \times \) determinant gives the original matrix.

Answer 1

Given the matrix \( B = \begin{bmatrix} 3 & a & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \)

Step 1: Relationship between adjoint and determinant The adjoint of a matrix \( A \), denoted as \(\text{adjoint}(A)\), satisfies \( A \cdot \text{adjoint}(A) = |A| \cdot I \), where \( I \) is the identity matrix and \( |A| \) is the determinant of \( A \). With \( |A| = 4 \), the relationship becomes \( A \cdot B = 4 \cdot I \) assuming \( B \) is the adjoint of \( A \).

Step 2: Properties of the adjoint matrix The adjoint matrix is the transpose of the cofactor matrix. For \( B \) to be the adjoint of \( A \), its entries must satisfy this property when multiplied with \( A \).

Step 3: Consistency check for \( \text{adjoint}(A) \) Since matrix \( B \) is symmetric, it is assumed to represent the adjoint matrix. For this to be consistent with \( |A| = 4 \), the diagonal entries of \( B \) must correspond to the cofactors of \( A \), and the off-diagonal entries must not negatively impact the determinant calculation. The symmetry of \( B \) implies that \( a = 1 \) is required for consistency.

Conclusion: The value of \( a \) is:

\[ \boxed{1} \] ---