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

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

(iii) If is invertible, the equation has the unique solution .

The function `inv(A)` computes the inverse of a matrix `A`,
if
is invertible.
Then the solution to
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 and . Then show that .

EXAMPLES 2. Find the inverse of

EXAMPLES 3. Solve the system

Department of Mathematics