If \( A=[a_{ij}] \) is a \( 3\times 3 \) matrix, whose elements are given by \( a_{ij}=i-j \), then \( A \) is a
(A) Symmetric matrix
(B) Skew symmetric matrix
(C) Square matrix
(D) Diagonal matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
Writing the matrix from \( a_{ij}=i-j \):
\( \displaystyle A=\begin{bmatrix}0 & -1 & -2\\1 & 0 & -1\\2 & 1 & 0\end{bmatrix} \)
In general
\( \displaystyle a_{ji}=j-i \)
\( \displaystyle =-(i-j)=-a_{ij} \)
So A is a 3 × 3 matrix with \( a_{ji}=-a_{ij} \), that is \( A^{\mathrm{T}}=-A \), and with non-zero entries off the diagonal.
Checking the statements
(A) Symmetric matrix — Incorrect. \( a_{12}=-1 \) but \( a_{21}=1 \).
(B) Skew symmetric matrix — Correct. \( a_{ji}=-a_{ij} \) for all i, j, so \( A^{\mathrm{T}}=-A \).
(C) Square matrix — Correct. It has 3 rows and 3 columns.
(D) Diagonal matrix — Incorrect. The entries off the diagonal, such as \( a_{12}=-1 \), are not zero.