Calculate the product ST, and infer from it the inverse of T.

Question

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.

Answer 1

HINT : So $ST$ is almost the identity matrix meaning that $S$ is almost the inverse of $T$.

Which calculations produces that $2$? How can you change $S$ such that this $2$ becomes $0$ and the rest remains unchanged?

Change it and then calculate again to check it didn't modify any other entries.

Answer 2

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 $$ \pmatrix{ 1&0&-2\\ 0&1&0\\ 0&0&1 } \;(ST) = \left(\pmatrix{ 1&0&-2\\ 0&1&0\\ 0&0&1 } \;S \right)\cdot T = I $$