Question

If \[ A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \] then $A$ is a/an:

• A. scalar matrix
• B. identity matrix
• C. symmetric matrix
• D. skew-symmetric matrix

Hint:

A scalar matrix must have equal diagonal elements, and all off-diagonal elements must be zero.

Answer 1

The Correct Option is A

A scalar matrix is a matrix in which all the diagonal elements are equal, and all the off-diagonal elements are zero.
The given matrix $A$ is: \[ A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \] This matrix is not a scalar matrix because its diagonal elements are not all equal. Thus, we rule out the scalar matrix option.
Next, we observe that a symmetric matrix is one where $A = A^T$, and a skew-symmetric matrix is one where $A = -A^T$. 
The given matrix $A$ is neither symmetric nor skew-symmetric, as it is not equal to its transpose and also not equal to the negative of its transpose. Therefore, the correct answer is that $A$ is a scalar matrix.