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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
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B
symmetric marix
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C
skew symmetric matrix
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D
orthogonal matrix
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Solution

The correct option is C skew symmetric matrix
Given: 
$$A = [a_{ij}]_{(3\times3)}$$

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

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

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

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

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

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

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

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


$$\therefore A=\begin{bmatrix}
0 &-3  &-8 \\
3 &  0&-5 \\
 8& 5 &0
\end{bmatrix}$$

Here, $$A^T=-A$$

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

Hence, option C.

Mathematics

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