Which of the following is/are correct?

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If A is a $n \times n$ matrix such that $a_{i j} = (i^{2} + j^{2} - 5 i j) \cdot (j - i) \forall i a n d j t h e n t r a c e (A) = 0$

If A is a $n \times n$ matrix such that $a_{i j} = (i^{2} + j^{2} - 5 i j) \cdot (j - i) \forall i a n d j t h e n t r a c e (A) \neq 0$

If P is a $3 \times 3$ orthogonal matrix $α, β, γ$ , are the angles made by a straight line with OX,OY,OZ and $A = [\begin{matrix} \sin^{2} α & \sin α \cdot \sin β & \sin α \cdot \sin γ \\ \sin α \cdot \sin β & \sin^{2} β & \sin β \cdot \sin γ \\ \sin α \cdot \sin γ & \sin β \cdot \sin γ & \sin^{2} γ \end{matrix}]$ and $Q = P^{T} A P,$ then $P Q^{6} P^{T} = 32 A$

If matrix $A = {[a_{i j}]}_{3 \times 3}$ and matrix $B = {[b_{i j}]}_{3 \times 3}$ where $a_{i j} + a_{j i} = 0$ and $b_{i j} - b_{j i} = 0$ $\forall i a n d j t h e n A^{6} B^{7}$ is a singular matrix

answer is A, C, D.

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Detailed Solution

A. $a_{i j} = - a_{j i} \Rightarrow a_{i j} + a_{i j} = 0 \Rightarrow$ A is skew symmetric matrix
C. $A^{2} = 2 A \Rightarrow A^{3} = 2^{2} A \Rightarrow A^{6} = 2^{5} A$
D. $A^{6} B^{7}$ is skew symmetric matrix of odd order