If \( A=[a_{ij}]_{3\times3} \) and \( a_{ij}=i-j \), then which of the following statements are TRUE?
(A). \( A \) is scalar matrix
(B). \( A \) is skew symmetric matrix
(C). \( |A|=0 \)
(D). \( a_{21}+a_{12}=0 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
With \( a_{ij}=i-j \),
\( \displaystyle A=\begin{bmatrix}0&-1&-2\\1&0&-1\\2&1&0\end{bmatrix} \)
\( a_{ji}=j-i=-a_{ij} \)
So A is a skew-symmetric matrix of odd order 3, its off-diagonal entries are not zero, and \( a_{21}=1,\ a_{12}=-1 \).
Checking the statements
(A) \( A \) is scalar matrix — Incorrect. A scalar matrix has zero off-diagonal entries, but here \( a_{12}=-1 \).
(B) \( A \) is skew symmetric matrix — Correct. \( a_{ji}=-a_{ij} \) for all i, j.
(C) \( |A|=0 \) — Correct. A skew-symmetric matrix of odd order has determinant 0.
(D) \( a_{21}+a_{12}=0 \) — Correct. \( 1+(-1)=0 \).