Question

If \[ A = \begin{bmatrix} 0 & 0 & -5 \\ 0 & 3 & 0 \\ 4.3 & 0 & 0 \end{bmatrix}, \text{ then A is a:} \]

• A. skew-symmetric matrix
• B. scalar matrix
• C. diagonal matrix
• D. square matrix

Hint:

A matrix is called a square matrix if the number of rows is equal to the number of columns, regardless of the elements inside.

Answer 1

The Correct Option is D

Let us examine the given matrix: \[ A = \begin{bmatrix} 0 & 0 & -5 \\ 0 & 3 & 0 \\ 4.3 & 0 & 0 \end{bmatrix} \] 1. It is a 3 × 3 matrix, i.e., same number of rows and columns. So, it's a square matrix.
2. Skew-symmetric matrix requires that $A^T = -A$ and all diagonal elements must be zero. But here, the $(2,2)$ entry is 3 $\ne 0$, so it is not skew-symmetric.
3. Scalar matrix requires all diagonal elements to be equal and all off-diagonal elements to be zero. Clearly, this is not the case here.
4. Diagonal matrix has non-zero elements only on the main diagonal. But here, $A_{1,3} = -5$ and $A_{3,1} = 4.3$ which are non-diagonal positions. So it's not a diagonal matrix.
Thus, the only correct classification is that it's a square matrix.