Question

If A is a square matrix for which $a_{ij} = i^{2} - j^{2}$, then A
A. Zero matrix
B. Unit matrix
C. Symmetric matrix
D. Skew symmetric matrix

Answer 1

Hint: Given a square matrix A and its element representation we need to identify the type of matrix. To identify the type we need to compare the elements of A with each other. Also, recall the definition for each of the above-mentioned matrices.

Complete step by step solution: We have a square matrix A with $a_{ij} = i^{2} - j^{2}$.
Now for this matrix to be a zero matrix all the elements must be zero, but this is not possible here because $a_{ij} = 0$ only when $i=j$since $a_{ii} =i^{2}-i^{2} = 0$. Therefore, only the diagonal elements will be zero, which implies A is not a zero matrix.
Clearly, A cannot be a unit matrix since the diagonal elements are zero.
Now, to find whether A is symmetric or skew-symmetric we need to compare $a_{ij}$ and $a_{ji}$.
$a_{ji} = j^{2}-i^{2} = -(i^{2}-j^{2}) = - a_{ij}$
So, if A is a $2\times 2$ matrix and $A = \begin{bmatrix}a_{11} & a_{12}\\ a_{21} & a_{22}\end{bmatrix}$ by applying the given condition we get,
$A= \begin{bmatrix}0 & a_{12}\\ -a_{12} & 0\end{bmatrix}$
then, $A^{T} = \begin{bmatrix}0 & -a_{12} \\ a_{12} & 0\end{bmatrix} = -\begin{bmatrix}0 & a_{12}\\ -a_{12} & 0\end{bmatrix} = -A$
Clearly, A is skew-symmetric.

So, Option ‘D’ is correct

Note: If we need to identify whether the given matrix is symmetric or skew-symmetric, check the value of diagonal elements. If the diagonal elements are zero, then the matrix is skew-symmetric. If a matrix is symmetric then either side of the diagonal elements will be the same.