Is this matrix exponentiation method correct?

[Jhenrique]

Is correct my step by step below?

$$\begin{aligned} \exp \left (\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} \right ) &= \exp \left ( a_{11} \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} + a_{22} \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix} + a_{12} \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} + a_{21} \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \right ) \\ & = \exp\left ( \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \right )^{a_{11}} \exp\left ( \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix} \right )^{a_{22}} \exp\left ( \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \right )^{a_{12}} \exp\left ( \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \right )^{a_{21}} \\ &= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{a_{11}} \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}^{a_{22}} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^{a_{12}} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}^{a_{21}} \end{aligned}$$

 

[tiny-tim]

Hi Jhenrique!

Hint:

i] what is exp$\begin{bmatrix} a & 0 \\ 0 & a \\ \end{bmatrix}$ ? what is exp$\begin{bmatrix} 0 & a \\ a & 0 \\ \end{bmatrix}$ ?

ii] do your final matrices commute?

 

[D H]

Jhenrique, I fixed your LaTeX so it doesn't spill across the screen.

The answer to your question is no. You made multiple errors. Your second line erroneously assumes $\exp(A+B) = \exp(A)\exp(B)$ and also erroneously assumes $\exp(sA)=\exp(A)^s$.