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:

In a scalar matrix, since all off-diagonal elements are zero, the product of all elements will always be zero.

Answer 1

The Correct Option is A

Step 1: Definition of Scalar Matrix.
A scalar matrix is a special type of diagonal matrix where every element on the diagonal is the same scalar, and all the off-diagonal elements are zeros. For a \( 3 \times 3 \) scalar matrix, the general structure is: \[ A = \begin{bmatrix} k & 0 & 0
0 & k & 0
0 & 0 & k \end{bmatrix}, \] where \( k \) represents the scalar value along the diagonal. Step 2: Summing the Elements.
The sum of all elements in a \( 3 \times 3 \) scalar matrix can be calculated by adding the diagonal elements. Given that the total sum is \( 9 \), we have: \[ \text{Sum} = k + k + k + 0 + 0 + 0 + 0 + 0 + 0 = 3k. \] From the equation, we know: \[ 3k = 9 \quad \Rightarrow \quad k = 3. \] Step 3: Product of All Elements.
Since the matrix is a scalar matrix, the product of all its elements involves multiplying the scalar \( k \) with the off-diagonal zeros: \[ \text{Product} = k \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 \cdot 0 = 0. \] Step 4: Final Answer.
Thus, the product of all elements in the matrix is: \[ \boxed{0}. \]