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, the product of all elements is always \( 0 \) due to the presence of off-diagonal zero elements.

Answer 1

The Correct Option is A

Step 1: Definition of a scalar matrix.
A scalar matrix is a diagonal matrix where all the diagonal elements are equal, and all off-diagonal elements are \( 0 \). For a \( 3 \times 3 \) scalar matrix, the general form is: \[ A = \begin{bmatrix} k & 0 & 0
0 & k & 0
0 & 0 & k \end{bmatrix}, \] where \( k \) is the scalar value on the diagonal. Step 2: Compute the sum of all elements.
The given sum of all elements in the matrix is \( 9 \). The sum of all elements of a \( 3 \times 3 \) scalar matrix is: \[ \text{Sum} = k + k + k + 0 + 0 + 0 + 0 + 0 + 0 = 3k. \] From the problem, we know: \[ 3k = 9 \quad \Rightarrow \quad k = 3. \] Step 3: Compute the product of all elements.
In a scalar matrix, all off-diagonal elements are \( 0 \). Therefore, the product of all elements in the matrix is: \[ \text{Product} = k \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 = 0. \] Step 4: Conclusion.
The product of all elements of the matrix is: \[ \boxed{0}. \]