A = [3x3] Matrix =
-1 3 -1
2 -2 3
-1 1 2
How Do We Get (A)^(-1) (The inverse of A)?
I would show an attempt at the solution, however, I am clueless about this.
Try this: append the matrix $\displaystyle I$ to $\displaystyle A$ thus:
$\displaystyle [A|I]=\left[
\begin{array}{rrr|rrr}
-1 &3 &-1 &1 &0 &0\\
2 &-2 &3 &0 &1 &0\\
-1 &1 &2 &0 &0 &1
\end{array}\right].$
Now perform on this entire matrix the row operations necessary to get to reduced row echelon form; that is, you take $\displaystyle A\to I.$ Then you'll have
$\displaystyle [A|I]\to[I|A^{-1}].$
All of this assumes, of course, that $\displaystyle A$ is invertible.
Make sense?