Question

Let A be a 3 × 3 matrix such that \(\text{det}(A) = 5\). If \(\text{det}(3 \, \text{adj}(2A)) = 2^{\alpha \cdot 3^{\beta} \cdot 5^{\gamma}}\), then \( (\alpha + \beta + \gamma) \) is equal to:

• A. 25
• B. 27
• C. 26
• D. 28

Hint:

For matrix determinant properties, remember that the adjoint's determinant is related to the original matrix's determinant raised to a power.

Answer 1

The Correct Option is B

The determinant of the adjoint of a matrix \(A\) is given by \(\text{det(adj}(A)) = (\text{det}(A))^{n-1}\) for an \(n \times n\) matrix. For a 3×3 matrix, this simplifies to \(\text{det(adj}(A)) = (\text{det}(A))^2\). Additionally, \(\text{det}(kA) = k^n \cdot \text{det}(A)\) for an \(n \times n\) matrix. Applying these properties, we calculate \(\text{det}(3 \, \text{adj}(2A)) = 3^3 \cdot 2^3 \cdot (\text{det}(A))^2\). Given \(\text{det}(A) = 5\), this evaluates to \(27 \cdot 8 \cdot 5^2\), which can be expressed as \(2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}\). This implies \(\alpha = 3\), \(\beta = 3\), and \(\gamma = 4\). Therefore, \(\alpha + \beta + \gamma = 3 + 3 + 4 = 27\).