If `A=[a_(i j)]` is a square matrix of even order such that `a_(i j)=i^2-j^2` , then (a) `A` is a skew-symmetric matrix and `|A|=0` (b) `A` is symmetric matrix and `|A|` is a square (c) `A` is symmetric matrix and `|A|=0` (d) none of these

Options

A. A is a skew-symmetric matrix and `absA=0`
B. A is symmetric matrix and `absA` is a square
C. A is symmetric matrix and `absA=0`
D. none of these

Answer

D

Explanation

We have, `a_(ij)=i^2-j^2`
`:. A_(ij)=j^2-i^2rArr a_(ij)=-a_(ij)`
Thus, A is a skew-summetric matrix of even order.
We know that the determinant of every skew-symmetric matrix of even order is a perfact square and that of odd order is zero.
Hence, option (d) is correct.