When computing the matrix \begin{bmatrix}1 & -\sqrt{3} \\ \sqrt{3} & 1\end{bmatrix} times its inverse, Wolfram Alpha produces this incorrect result. The same result happens when Power is changed to MatrixPower. I suspect that this is a parsing problem, that somehow Power/MatrixPower is interpreted as something else. Normally, negative power is interpreted as inverses according to the documentation center. Likewsie, if I change Power to Inverse, it also produces the correct result. As far as I know, it is also not taking the inverse of each individual elements in the matrix, and adding parathesese or brackets around the first term does not change the result.
I have also tried some other examples: example 1, example 2. What exactly is Wolfram Alpha treating MatrixPower($m,-1$) as?
It's computing the inverse correctly, but doing the product elementwise. Try power[{{1,-Sqrt[3]},{Sqrt[3],1}},-1] . {{1,-Sqrt[3]},{Sqrt[3],1}}.