If the matrix \( A=\begin{bmatrix}-2 & 0 & 0\\0 & 3 & 0\\0 & 0 & 4\end{bmatrix} \) is given, then which of the following statements is/are TRUE?
A. \( A \) is a square matrix
B. \( A \) is a symmetric matrix
C. \( A \) is a scalar matrix
D. \( A \) is a identity matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A and B only
A has 3 rows and 3 columns, and every entry off the main diagonal is 0, so A is a diagonal matrix.
A scalar matrix needs all diagonal entries equal, and the identity matrix needs them all equal to 1.
So A is a 3 × 3 diagonal matrix whose diagonal entries −2, 3 and 4 are neither all equal nor all 1.
Checking the statements
(A) \( A \) is a square matrix — Correct. It has 3 rows and 3 columns.
(B) \( A \) is a symmetric matrix — Correct. Every off-diagonal entry is 0, so \( a_{ij}=a_{ji} \) and \( A'=A \).
(C) \( A \) is a scalar matrix — Incorrect. The diagonal entries −2, 3 and 4 are not equal.
(D) \( A \) is a identity matrix — Incorrect. The diagonal entries are −2, 3 and 4, not all 1.