Question

If \( A = \text{diag}(d_1, d_2, \ldots, d_n) \), then \( |A| \) is:

• A. 0
• B. \( d_1 + d_2 + \cdots + d_n \)
• C. \( d_1 d_2 \cdots d_n \)
• D. None of these

Hint:

The determinant of a diagonal matrix is the product of the diagonal elements.

Answer 1

The Correct Option is C

Step 1: Understanding the diagonal matrix.
A diagonal matrix is a square matrix in which the elements outside the main diagonal are zero. The determinant of a diagonal matrix is the product of the elements on the main diagonal. Step 2: Applying the determinant formula.
For the diagonal matrix \( A = \text{diag}(d_1, d_2, \ldots, d_n) \), the determinant is: \[ |A| = d_1 \cdot d_2 \cdot \cdots \cdot d_n \] Step 3: Conclusion.
Thus, the determinant of the matrix \( A \) is the product of its diagonal elements, corresponding to option (c).