If $A = [a_{ij}]$ be square matrix of order 3, such that $a_{ij} = i + j, ∀i,j$ 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
Given: $A = [a_{ij}]$ of order 3 with $a_{ij} = i + j$
Matrix A:
$A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix}$
Check properties:
(A) Skew-symmetric: $A^T = -A$ → Not true ($A^T = A$), so False
(B) Non-singular: Check determinant:
$\det(A) = \begin{vmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{vmatrix} = 2(24-25) - 3(18-20) +4(15-16) = 2(-1)-3(-2)+4(-1)=-2+6-4=0$
So singular → inverse does not exist, hence B is False, C is True
(D) Symmetric: $A^T = A$ → True