If \(A = [a_{ij}]\) be square matrix of order 3, such that \(a_{ij} = i + j,\ \forall\, i, j\) then which of the following are correct?
(A) A is a skew-symmetric matrix.
(B) A is a non-singular matrix.
(C) The inverse of A does not exist.
(D) A is a symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (C) and (D) only
From \(a_{ij} = i + j\):
\( A = \begin{bmatrix}2&3&4\\3&4&5\\4&5&6\end{bmatrix} \)
Applying \(C_3 \rightarrow C_3 - C_2\) and then \(C_2 \rightarrow C_2 - C_1\):
\( |A| = \begin{vmatrix}2&1&1\\3&1&1\\4&1&1\end{vmatrix} = 0 \)
since two columns are identical.
So A is symmetric, with diagonal entries 2, 4, 6, and \(|A| = 0\).
Checking the statements
(A) A is a skew-symmetric matrix. — Incorrect. Its diagonal entries \(a_{ii} = 2i\) are not 0.
(B) A is a non-singular matrix. — Incorrect. \(|A| = 0\), so A is singular.
(C) The inverse of A does not exist. — Correct. A is singular, so \(A^{-1}\) does not exist.
(D) A is a symmetric matrix. — Correct. \(a_{ji} = j + i = a_{ij}\) for all i, j.