Question

If $A=[a_{ij}{]}_{3\times 3}$ is a square matrix so that $a_{ij}=i^2-j^2$, then $A$ is a

• A. unit matrix
• B. symmetric marix
• C. skew symmetric matrix
• D. orthogonal matrix

Answer 1

The correct option is C skew symmetric matrix

Given:

$A=[a_{ij}{]}_{(3\times 3)}$

where, $a_{ij}=i^2-j^2$

$\therefore a_{ij}=0$ if $i=j$

Now,

$a_{12}=1^2-2^2=-3$

$a_{13}=1^2-3^2=-8$

$a_{21}=2^2-1^2=3$

$a_{23}=2^2-3^2=-5$

$a_{31}=3^2-1^2=8$

$a_{32}=3^2-2^2=5$

$\therefore A=\left[\begin{array}{ccc}0& -3& -8\\ 3& 0& -5\\ 8& 5& 0\end{array}\right]$

Here, $A^T=-A$

$\therefore A$ is a skew-symmetric matrix.

Hence, option C.