I am reading through Axler Linear Algebra Done Right and come across this example.
Define $T\in\mathcal{L}(\mathbb{R}^2)$ by $$T(x,y)=(41x+7y,-20x+74y).$$
The matrix of $T$ with respect to the standard basis of $\mathbb{R}^2$ is $$\begin{pmatrix}41&7\\-20&74\end{pmatrix},$$ which is not a diagonal matrix. However, $T$ is diagonalizable, because the matrix of $T$ with respect to the basis $(1,4),(7,5)$ is $$\begin{pmatrix}69&0\\0&46\end{pmatrix},$$ as you should verify.
Well, I am trying to verify it, but I keep getting the matrix of $T$ with respect to that latter basis as $$\begin{pmatrix}69&322\\276&230\end{pmatrix}.$$
Am I doing something wrong, or is the book in error? More generally, how can I find a diagonalization of a matrix whenever I want to?