Inverse of a matrix

Definition

We say that $B$ is an inverse of $A$ if

$$AB=BA=I$$

and it is denoted as $A^{-1}$ if it exists.

Check Invertibility of a matrix for the existence of $A^{-1}$

Inverse of matrix is unique, if it exists

Suppose that both $B_1,B_2$ are inverses of $A$. Then

$$B_1=B_{1}I=B_1(A{B}_2)=(B_{1}A)B_2=I{B}_2=B_2$$

$$A^{-1}=\frac{1}{\det A}adj(A)$$

where $adj(A)$ is Adjugate Matrix of $A$.

Gauss-Jordan method

Another approach to invert a matrix is to use Gaussian elimination

1. Obtain $[A|I]$
2. Apply elimination until you reach $[I|A^{-1}]$.