If \( A=[a_{ij}]_{3\times 3},\ a_{ij}=\begin{cases}0, & i\ne j\\i+j, & i=j\end{cases} \), then which of the following statements are correct ?
A. \( A \) is scalar matrix
B. \( A \) is diagonal matrix
C. \( A \) is unit matrix
D. \( A \) is symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and D only
The off-diagonal entries are 0 and the diagonal entries are \( a_{ii}=i+i=2i \):
\( \displaystyle A=\begin{bmatrix}2 & 0 & 0\\0 & 4 & 0\\0 & 0 & 6\end{bmatrix} \)
So A is a diagonal matrix with diagonal entries 2, 4 and 6.
Checking the statements
(A) \( A \) is scalar matrix — Incorrect. A scalar matrix has all diagonal entries equal; here they are 2, 4, 6.
(B) \( A \) is diagonal matrix — Correct. Every off-diagonal entry is 0.
(C) \( A \) is unit matrix — Incorrect. The diagonal entries are 2, 4, 6, not all 1.
(D) \( A \) is symmetric matrix — Correct. Every diagonal matrix satisfies \( A^T=A \).