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

Solution

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