If A = [aij] is a square matrix such that aij = i2 – j2, then write whether A is symmetric or skew-symmetric.
Given: A = [aij] is a square matrix such that aij = i2 – j2
Suppose A is a 2 × 2 square matrix i.e.
Here,
aij = i2 – j2
So, a12 = (1)2 – (2)2 = 1 – 4 = - 3
and a21 = (2)2 – (1)2 = 4 – 1 = 3
For diagonal elements, i = j, we have
a11 = (1)2 – (1)2 = 0
and a22 = (2)2 – (2)2 = 0
So, Matrix A becomes
Now, we have to check A is symmetric or skew – symmetric.
We know that, if a matrix is symmetric then AT = A
and if a matrix is skew – symmetric then AT = -A
So, firstly we find the AT
So,
⇒ AT = - A
∴ A is a skew – symmetric matrix.