linear algebra – The matrix of \$T(x_1,x_2)=(x_1+x_2,x_1-x_2)\$ with respect to a basis

Consider the linear map $$T:mathbb{R}^2to mathbb{R}^2, T(x_1, x_2)=(x_1+x_2,x_1-x_2)$$. Let $$B_1$$ be the canonical base of $$mathbb{R}^2$$ and consider another basis $$B_2={f_1,f_2}$$, where $$f_1=(1,1)$$ and $$f_2=(1,2)$$.
So, according to my computations, the matrix of $$T$$ with respect to $$B_1$$ is $$begin{pmatrix} 1 & 1\ 1 & -1 end{pmatrix}$$ and the matrix of $$T$$ with respect to $$B_2$$ is $$begin{pmatrix} 2 & 3 \ 0 & -1 end{pmatrix}$$. However, I tried computing the transition matrices. I got that the transition matrix from $$B_1$$ to $$B_2$$ is $$begin{pmatrix} 1 & 1\ 1 & 2 end{pmatrix}$$ and the transition matrix from $$B_2$$ to $$B_1$$ is $$begin{pmatrix} 2 & -1\ -1 & 1 end{pmatrix}$$. I should have that $$begin{pmatrix} 2 & 3 \ 0 & -1 end{pmatrix}= begin{pmatrix} 2 & -1\ -1 & 1 end{pmatrix}cdot begin{pmatrix} 1 & 1\ 1 & -1 end{pmatrix}cdot begin{pmatrix} 1 & 1\ 1 & 2 end{pmatrix}$$, but this is not true. Where did I go wrong?