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 has identical diagonal elements and zero off-diagonal elements. The provided matrix $A$ is: \[ A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \] This matrix is not scalar, as its diagonal elements (-1, 3, 5) are not equal. Therefore, it is not a scalar matrix. A symmetric matrix satisfies $A = A^T$, while a skew-symmetric matrix satisfies $A = -A^T$. The given matrix $A$ is neither symmetric nor skew-symmetric because it does not equal its transpose, nor does it equal the negative of its transpose. Consequently, the correct classification for $A$ is a scalar matrix.