If-A-is-a-square-matrix-of-order-3-then-A-A-T-2011-

Question Number 41321 by rahul 19 last updated on 05/Aug/18

If A is a square matrix of order 3, then

∣ {(A - A^{T})}^{2011} ∣ = ?

Commented by rahul 19 last updated on 05/Aug/18

I^{'} m getting A - A^{T} a skew symmetric

of order 3 .

I wanted to know why {(A - A^{T})}^{2011} will

also be a skew synmetric of order 3!

Answered by MJS last updated on 05/Aug/18

(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) - (\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}) = (\begin{matrix} 0 & p & q \\ - p & 0 & r \\ - q & - r & 0 \end{matrix})

{(\begin{matrix} 0 & p & q \\ - p & 0 & r \\ - q & - r & 0 \end{matrix})}^{2 n - 1} =

= (\begin{matrix} 0 & {(- 1)}^{n - 1} p {(p^{2} + q^{2} + r^{2})}^{n - 1} & {(- 1)}^{n - 1} q {(p^{2} + q^{2} + r^{2})}^{n - 1} \\ {(- 1)}^{n} p {(p^{2} + q^{2} + r^{2})}^{n - 1} & 0 & {(- 1)}^{n - 1} r {(p^{2} + q^{2} + r^{2})}^{n - 1} \\ {(- 1)}^{n} q {(p^{2} + q^{2} + r^{2})}^{n - 1} & {(- 1)}^{n} r {(p^{2} + q^{2} + r^{2})}^{n - 1} & 0 \end{matrix})

Commented by rahul 19 last updated on 06/Aug/18

thanks sir ��