Let \(A = [a_{ij}]\) be a square matrix of order 3 whose elements are given by \(a_{ij} = i - j\). Consider the following statements:
(A) A is a skew-symmetric matrix.
(B) The sum of the diagonal elements of A is 0.
(C) \(a_{31} = -2\)
(D) \(a_{23} = 1\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (B) only
Putting \(i = 1, 2, 3\) and \(j = 1, 2, 3\) in \(a_{ij} = i - j\):
\( \displaystyle A = \begin{bmatrix}0&-1&-2\\1&0&-1\\2&1&0\end{bmatrix} \)
Also \(a_{ji} = j - i = -(i - j) = -a_{ij}\) for every i and j.
So \(A' = -A\), every diagonal element is 0, \(a_{31} = 2\) and \(a_{23} = -1\).
Checking the statements
(A) A is a skew-symmetric matrix. — Correct. \(a_{ji} = -a_{ij}\), so \(A' = -A\).
(B) The sum of the diagonal elements of A is 0. — Correct. \(a_{ii} = i - i = 0\), so the sum is \(0 + 0 + 0 = 0\).
(C) \(a_{31} = -2\) — Incorrect. \(a_{31} = 3 - 1 = 2\).
(D) \(a_{23} = 1\) — Incorrect. \(a_{23} = 2 - 3 = -1\).