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:
- The inverse exists if and only if elimination produces n pivots (row exchanges are allowed).
- The matxi A cannot have two different inverses.
- If A is invertible, the one and only solution to Ax = b is $x = A^{-1}b$.
- (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 $x = A^{-1}\cdot 0 = 0$.
- 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 - A diagonal matrix has an inverse provided no diagonal entries are zero:
if $$ A = \begin{bmatrix} d_1 & & \\ &\ddots& \\ & &d_n\\ \end{bmatrix} $$ then $$ A^{-1} = \begin{bmatrix} \frac{1}{d_1} & & \\ &\ddots& \\ & &\frac{1}{d_n}\\ \end{bmatrix} $$