If A is a symmetric matrix and \(n \in \mathrm{N}\), then \(\mathrm{A}^{n}\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Symmetric matrix
A is symmetric means \(A^{T} = A\). Using \((AB)^{T} = B^{T}A^{T}\) repeatedly, the transpose of a power is the power of the transpose:
\(\left(A^{n}\right)^{T} = \left(A^{T}\right)^{n}\)
\(\left(A^{n}\right)^{T} = A^{n}\)
So \(A^{n}\) is also symmetric.