Let \( A=[a_{ij}]_{3\times 3} \), where \( a_{ij}=i-j \) , then which of the following statements is/are TRUE?
A. \( A \) is diagonal matrix
B. \( A \) is square matrix
C. \( A \) is symmetric matrix
D. \( A \) is skew symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B and D only
With \( a_{ij}=i-j \), the matrix is
\( \displaystyle A=\begin{bmatrix}0 & -1 & -2\\1 & 0 & -1\\2 & 1 & 0\end{bmatrix} \)
\( \displaystyle a_{ji}=j-i=-a_{ij} \)
So A is a 3 × 3 matrix with non-zero off-diagonal entries and \( a_{ji}=-a_{ij} \) for all i, j.
Checking the statements
(A) \( A \) is diagonal matrix — Incorrect. The off-diagonal entries are not zero, for example \( a_{21}=1 \).
(B) \( A \) is square matrix — Correct. It is of order 3 × 3.
(C) \( A \) is symmetric matrix — Incorrect. \( a_{12}=-1 \) but \( a_{21}=1 \).
(D) \( A \) is skew symmetric matrix — Correct. \( a_{ji}=-a_{ij} \) for all i, j, so \( A'=-A \).