\( A \) is any matrix, defined by \( A=[a_{ij}]_{3\times 3} \), where \( a_{ij}=|i-j| \), then which of the following statements are true ?
A. \( A \) is symmetric matrix.
B. \( A \) is skew-symmetric matrix.
C. \( A \) is diagonal matrix.
D. \( A \) is non singular matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A and D only
With \( a_{ij}=|i-j| \), the matrix is
\( \displaystyle A=\begin{bmatrix}0 & 1 & 2\\1 & 0 & 1\\2 & 1 & 0\end{bmatrix} \)
Expanding along the first row,
\( \displaystyle |A|=0(0-1)-1(0-2)+2(1-0) \)
\( \displaystyle =0+2+2=4 \)
So \( a_{ij}=a_{ji} \) for all i, j, the off-diagonal entries are not zero, and |A| = 4.
Checking the statements
(A) \( A \) is symmetric matrix. — Correct. \( |i-j|=|j-i| \), so \( a_{ij}=a_{ji} \).
(B) \( A \) is skew-symmetric matrix. — Incorrect. Skew-symmetry needs \( a_{ji}=-a_{ij} \), but \( a_{12}=a_{21}=1 \).
(C) \( A \) is diagonal matrix. — Incorrect. The off-diagonal entries are not zero, for example \( a_{12}=1 \).
(D) \( A \) is non singular matrix. — Correct. |A| = 4 ≠ 0.