Properties of $ A^{-1}$

(i) If a square matrix $ A$ satisfies ``either $ A C = I_n$ or $ C A = I_n$,'' then $ A$ is invertible, and has the unique inverse $ C$.

(ii) When $ A$ and $ B$ are invertible, we have the following properties:

$\displaystyle (A^{-1})^{-1} = A; \hspace{0.3in}
(AB)^{-1} = B^{-1} A^{-1}; \hspace{0.3in}
(A^T)^{-1} = (A^{-1})^T .
$

(iii) If $ A$ is invertible, the equation  $ A \mathbf{x} = \mathbf{b}$ has the unique solution  $ \mathbf{x} = A^{-1} \mathbf{b}$.

The function inv(A) computes the inverse of a matrix A, if $ \texttt{A}$ is invertible. Then the solution to  $ A \mathbf{x} = \mathbf{b}$ is given by inv(A) * b. Besides, Matlab/Octave has the special operator ``\'', called left division, which computes the solution immediately by A \ b. In summary the following two commands give the same solution:

> inv(A) * b
> A \ b

EXAMPLES 1. Let $ A =
\left[\begin{array}{cc}
2 & 5 \\
-3 &-7
\end{array}\right]$ and $ C =
\left[\begin{array}{cc}
-7 &-5 \\
3 & 2
\end{array}\right]$. Then show that $ C = A^{-1}$.

EXAMPLES 2. Find the inverse of $ A =
\left[\begin{array}{cc}
3 & 4 \\
5 & 6
\end{array}\right]$

EXAMPLES 3. Solve the system

\begin{displaymath}
\begin{array}{rrr}
3x_1 & + 4 x_2 & = 3 \\
5x_1 & + 6 x_2 & = 7
\end{array}\end{displaymath}



Department of Mathematics
Last modified: 2005-10-21