Question

Evaluate the determinant: \[ \left| \begin{matrix} 3 & \sqrt{3} & \sqrt{3} \\ 4 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right| \]

• A. 0
• B. 12
• C. \( 4\sqrt{3} \)
• D. \( 3 - 4\sqrt{3} \)

Hint:

When calculating the determinant of a matrix with rows or columns containing zeros, perform cofactor expansion along that row/column to simplify the calculation.

Answer 1

The Correct Option is A

We are given the matrix: \[ A = \begin{pmatrix} 3 & \sqrt{3} & \sqrt{3} \\ 4 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \] To evaluate the determinant of this 3x3 matrix, we use the cofactor expansion along the third row (because it contains two zeros, which simplifies the calculation). The cofactor expansion is: \[ \text{det}(A) = 0 \cdot \text{Cofactor of first element} - 0 \cdot \text{Cofactor of second element} + 0 \cdot \text{Cofactor of third element} \] Since all terms contain a factor of 0, the determinant is: \[ \text{det}(A) = 0 \] Thus, the correct answer is: (A) 0.