Exponentiation of a symmetric $2 \times 2$ matrix Why does the following hold?
$$\left[\begin{matrix}a & b\\b & a\end{matrix}\right]^k=\frac{1}{2}\left[\begin{matrix}\left(a - b\right)^{k} + \left(a + b\right)^{k} & - \left(a - b\right)^{k} + \left(a + b\right)^{k}\\- \left(a - b\right)^{k} + \left(a + b\right)^{k} & \left(a - b\right)^{k} + \left(a + b\right)^{k}\end{matrix}\right]$$

You can prove it by induction. The induction step would be 
\begin{align}
\begin{bmatrix}a&b\\ b&a\end{bmatrix}^{k+1}
&=\begin{bmatrix}a&b\\ b&a\end{bmatrix}\left[\begin{matrix}\left(a - b\right)^{k} + \left(a + b\right)^{k} & - \left(a - b\right)^{k} + \left(a + b\right)^{k}\\- \left(a - b\right)^{k} + \left(a + b\right)^{k} & \left(a - b\right)^{k} + \left(a + b\right)^{k}\end{matrix}\right]\\ \ \\
&=\left[\begin{matrix}\left(a - b\right)^{k+1} + \left(a + b\right)^{k+1} & - \left(a - b\right)^{k+1} + \left(a + b\right)^{k+1}\\- \left(a - b\right)^{k+1} + \left(a + b\right)^{k+1} & \left(a - b\right)^{k} + \left(a + b\right)^{k+1}\end{matrix}\right].
\end{align}
The computations are straightforward: for example for the 1,1 entry you have 
$$
a(a-b)^k+a(a+b)^k-b(a-b)^k+b(a+b)^k
=(a-b)(a-b)^k+(a+b)(a+b)^k
=(a-b)^{k+1}+(a+b)^{k+1}.
$$
