Question

If \( A = [a_{ij}]_{3 \times 3} \) is a scalar matrix then which of the following must be true ?

• A. \( A \) must be a symmetric matrix.
• B. \( A \) must be a skew-symmetric matrix.
• C. \( A \) must be an identity matrix.
• D. \( A \) must be a null matrix.

Hint:

Tip 1: Every diagonal matrix (and thus every scalar and identity matrix) is always a symmetric matrix.
Tip 2: The only scalar matrix that is also skew-symmetric is the Zero matrix.

Answer 1

The Correct Option is A

Concept:
• A scalar matrix is a diagonal matrix in which all the diagonal elements are equal.
• For a matrix \( A = [a_{ij}] \), it is a scalar matrix if \( a_{ij} = 0 \) for \( i \neq j \) and \( a_{ii} = k \) for some constant \( k \).
• A matrix is symmetric if \( A = A^T \), i.e., \( a_{ij} = a_{ji} \) for all \( i, j \).

Step 1: Write the general form of a \( 3 \times 3 \) scalar matrix
Let \( A \) be a scalar matrix with diagonal elements \( k \): \[ A = \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix} \]

Step 2: Find the transpose of the matrix
The transpose \( A^T \) is obtained by interchanging rows and columns: \[ A^T = \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix} \]

Step 3: Compare the matrix with its transpose
Since \( A = A^T \), the matrix is symmetric. Note that it is not necessarily an identity matrix (unless \( k=1 \)) or a null matrix (unless \( k=0 \)). It is not skew-symmetric because diagonal elements of a skew-symmetric matrix must be zero.