Matrix Algebra

Properties of Inverse

2-by-2 Inverse Matrix. Let $A = \begin{bmatrix}a & b c & d \end{bmatrix}$ be a $2 \times 2$ matrix. Then we can explicitly compute the inverse

$\displaystyle A^{-1} = \begin{bmatrix} a & b c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b -c & a \end{bmatrix}$

is invertible, that is, if $ad - bc \neq 0$ .

Properties of Inverse.

If an $n\times n$ matrix satisfies “either or ,” then is invertible, and has the unique inverse .
When and are invertible, we have the following properties:

$\displaystyle (A^{-1})^{-1} = A; \hspace{0.3in} (AB)^{-1} = B^{-1} A^{-1}; \hspace{0.3in} (A^T)^{-1} = (A^{-1})^T .$
If is invertible, the equation $A \mathbf{x} = \mathbf{b}$ has the unique solution $\mathbf{x} = A^{-1} \mathbf{b}$ .

Matlab/Octave. The function inv(A) computes the inverse of a matrix A, if 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. Find the inverse of $A = \left[\begin{array}{cc} 3 & 4 \\ 5 & 6 \end{array}\right]$ . Then solve the following system of linear equations.

$\begin{displaymath} \left\{ \begin{array}{rrr} 3x_1 & +\,4 x_2 & = 3 \\ 5x_1 & +\,6 x_2 & = 7 \end{array} \right. \end{displaymath}$