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Please Solve RD Sharma Class 12 Chapter 4 Algebra of Matrices Exercise Very Short Asnwer Question 28 Maths Textbook Solution.

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Answer: nis even SoA^{n} is symmetric

Hint: We must know the basic algebra

Given: Ais a skew symmetric matrix

Solution: Awhere n\epsilon A

A^{T}=-A    (Ais a skew symmetric matrix)

\begin{aligned} \left(A^{n}\right)^{T}=\left(A^{n-1} \times A^{1}\right)^{T} &=A^{T} \times\left(A^{n-1}\right)^{T} \quad\left[(A B)^{T}=B^{T} A^{T}\right] & \\ =(-A) \times\left(A^{n-2} \times A\right)^{T} &=(-A) A^{T}\left(A^{n-2}\right)^{T} &=(-A)(-A)\left(A^{n-3} \times A\right)^{T} \end{aligned}

As the same sequence will be continue ,so it can be written in general term as         =\left ( -1 \right )^{n}A^{n}=1A^{n}

So, n is even.

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