Something to note

The matrix A is invertible if there exists a matrix $A^{-1}$

Two-sided inverse: $$ A^{-1}\cdot A = I $$ Not all matrices have inverses. There are six note:

1. The inverse exists if and only if elimination produces n pivots (row exchanges are allowed).
2. The matxi A cannot have two different inverses.
3. If A is invertible, the one and only solution to Ax = b is $x = A^{-1}b$.
4. (important) Suppose tehre is a nonzero vector x such that Ax = 0. Then A cannot have an inverse. No matrix can bring o back to x.

   If A is invertible, then Ax = 0 can only have the zero solution when $x = A^{-1}\cdot 0 = 0$.

5. A 2 by 2 matrix is invertible if an only if ad-bc is not zero:

   2 by 2 Inverse: $$ \begin{bmatrix} a & b
   c & d
   \end{bmatrix} ^{-1}=\frac{1}{ad-bc} \begin{bmatrix} d & -b
   -c & a
   \end{bmatrix} $$
   ad-bc is the determinant of A

6. A diagonal matrix has an inverse provided no diagonal entries are zero:
   if $$ A = \begin{bmatrix} d_1 & &
   &\ddots&
   & &d_n
   \end{bmatrix} $$ then $$ A^{-1} = \begin{bmatrix} \frac{1}{d_1} & &
   &\ddots&
   & &\frac{1}{d_n}
   \end{bmatrix} $$