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If A is a symmetric matrix and n in N ,...

If `A` is a symmetric matrix and `n in N ,` write whether `A^n` is symmetric or skew-symmetric or neither of these two.

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If A is a symmetric matrix, then `A^{T}=A. `
Now,
` (A^{n})^{T}=(A^{T})^{n} quad [ text{for all } n in N] `
`Rightarrow(A^{n})^{T}=(A)^{n} [ text{because } A^{T}=A] `
Hence, is a symmetric matrix. `(A)^{n}`
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