Is $x$ raised to the power of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ equal to $\begin{bmatrix}x^a&x^b-1\\x^c-1&x^d\end{bmatrix}$?

Question

WolframAlpha says $x$ raised to the power of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is equal to $\begin{bmatrix}x^a&x^b\\x^c&x^d\end{bmatrix}$.

$x = e^{\ln x}$ , and ${x^a}^b = x^{a \cdot b}$
$\therefore$ $x^M = e^{\ln x \cdot M}$ , where $M$ is a matrix.

According to 3Blue1Brown, $e^{y \cdot M} = y^0 \cdot M^0 + \sum_{n=1}^{\infty}\left( \frac{y^n}{n!} \cdot M^n \right)$

But when I asked to Mathics (open source alternative to Mathematica) the result of the sum above, it said it's $\begin{bmatrix}x^a&x^b-1\\x^c-1&x^d\end{bmatrix}$

Which one is wrong?

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If anyone want to know what I typed in Mathics:

IdentityMatrix[2] + Sum[((Log[x]^n/n!) * {{a,b},{c,d}}^n), {n, 1, Infinity}]

(in Mathics Log[x] is $\ln x$)

Answer 1

In Mathematica and all things pretending to be Mathematica, x^{{a,b},{c,d}} and {{a,b},{c,d}}^x don't know that you're dealing with a matrix, and are going to apply the operation elementwise.

You want MatrixPower[{{a,b},{c,d}},n] to raise a matrix to the $n^{\text{th}}$ power, and MatrixFunction[x^#&, {{a,b},{c,d}}] or MatrixExp[Log[x]{{a,b},{c,d}}] for the matrix exponential with the weird base.