Question

If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements is:

• A. \( 0 \)
• B. \( 9 \)
• C. \( 27 \)
• D. \( 729 \)

Hint:

For scalar matrices, remember that only diagonal elements are non-zero. Off-diagonal elements contribute nothing to the product.

Answer 1

The Correct Option is A

Step 1: {Understand a scalar matrix}
A scalar matrix is a diagonal matrix where all diagonal elements are equal, and all off-diagonal elements are zero. Let the diagonal element be \( a \). In a \( 3 \times 3 \) matrix, the sum of elements is: \[ 3a = 9 \quad \Rightarrow \quad a = 3. \] Step 2: {Find the product of all elements}
The product of all elements of the scalar matrix is: \[ a^3 \cdot 0 = 0, \] as all off-diagonal elements are zero. Conclusion: The product of all elements is \( 0 \).