3.5 Invertible Matrices

3.5 Invertible Matrices

A square matrix $A$ of order $n$ is invertible if there exists a matrix $B$ such that:

$$AB = BA = I_n$$

Then $B = A^{-1}$.

Inverse of a $2 \times 2$ Matrix

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if $\det(A) = ad - bc \neq 0$:

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

Example

Find the inverse of $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.

$$\det(A) = 2(4) - 3(1) = 8 - 3 = 5$$ $$A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \\ -\frac{1}{5} & \frac{2}{5} \end{bmatrix}$$