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?