<s>[INST] Calculate the product ST, and infer from it the inverse of T. S=\begin{pmatrix}
1/2 & 1/2 & 0\\ 
1 & 0 & 0\\ 
-3/2 & 0 & 1/2
\end{pmatrix}
T=
\begin{pmatrix}
0 & 1 & 0\\ 
2 & -1 & 4\\ 
0 & 3 & 2
\end{pmatrix}
I have calculated ST to be =
\begin{pmatrix}
1 & 0 & 2\\ 
0 & 1 & 0\\ 
0 & 0 & 1
\end{pmatrix}
But i'm now unsure of how to continue? This is obviously a significant matrix but I can't spot the answer.
Thanks.
 [/INST] Think in terms of Gaussian elimination.  It should be clear that
$$
\pmatrix{
1&0&-2\\
0&1&0\\
0&0&1
}
\pmatrix{
1&0&2\\
0&1&0\\
0&0&1
} = I
$$
That is, we have
$$
\pmatrix{
1&0&-2\\
0&1&0\\
0&0&1
}
\;(ST) = 
\left(\pmatrix{
1&0&-2\\
0&1&0\\
0&0&1
}
\;S \right)\cdot T = I
$$
</s>