# Inverse Matrices

A matrix is invertible if its determinant is not 0. If A is an invertible matrix, then its inverse is $A^{-1}$ where $A^{-1}=\frac{1}{\left|A\right|} \cdot adj(A)$. $adj(A)$ is the adjugate of the matrix.

## Determining the Inverse of a Matrix

1. We calculate the determinant of the matrix.
2. We write the transpose of the matrix.
3. Every element of the transpose is replaced with its cofactor. The resulting matrix is the adjugate.
4. We calculate the inverse.

Example 46
$A=\begin{pmatrix} 1 & 3\\ 2 & 5 \end{pmatrix}$

$\left|A\right|=1\cdot 5-6=-1$
The matrix is invertible, so we can calculate its inverse.

$A^{T}= \begin{pmatrix} 1 & 2\\ 3 & 5 \end{pmatrix}$

We replace the elements of the transpose with their cofactors.

$1\longrightarrow (-1)^{1+1}\cdot \Delta_{1,1}=(-1)^{2}\cdot5 = 5$
$2\longrightarrow (-1)^{1+2}\cdot \Delta_{1,2}=(-1)^{3}\cdot3 = -3$
$3\longrightarrow (-1)^{2+1}\cdot \Delta_{2,1}=(-1)^{3}\cdot2 = -2$
$5\longrightarrow (-1)^{2+2}\cdot \Delta_{2,2}=(-1)^{4}\cdot1 = 1$

$adj(A)= \begin{pmatrix} 5 & -3\\ -2 & 1\\ \end{pmatrix}$

$A^{-1}=- \begin{pmatrix} 5 & -3\\ -2 & 1 \end{pmatrix} = \begin{pmatrix} -5 & 3\\ 2 & -1 \end{pmatrix}$

Example 47
$B=\begin{pmatrix} 2 & -7\\ -1 & 6 \end{pmatrix}$

$\left|B\right|=2\cdot 6-(-7)\cdot (-1) = 5$

The matrix invertible, so we can calculate its inverse.
$A^{T}= \begin{pmatrix} 2 & -1\\ -7 & 6 \end{pmatrix}$

We replace the elements of the transpose with their cofactors.
$2\longrightarrow (-1)^{1+1}\cdot \Delta_{1,1}=(-1)^{2}\cdot6 = 6$
$-1\longrightarrow (-1)^{1+2}\cdot \Delta_{1,2}=(-1)^{3}\cdot(-7) = 7$
$-7\longrightarrow (-1)^{2+1}\cdot \Delta_{2,1}=(-1)^{3}\cdot(-1) = 1$
$6\longrightarrow (-1)^{2+2}\cdot \Delta_{2,2}=(-1)^{4}\cdot2 = 2$

$adj(A)= \begin{pmatrix} 6 & 7\\ 1 & 2 \end{pmatrix}$

$A^{-1}=\frac{1}{5} \begin{pmatrix} 6 & 7\\ 1 & 2 \end{pmatrix} = \begin{pmatrix} \frac{6}{5} & \frac{7}{5}\\ \frac{1}{5} & \frac{2}{5} \end{pmatrix}$

Example 48
$C=\begin{pmatrix} 1 & 3 & 2\\ 4 & 1 & 1\\ 1 & 2 & 3\\ \end{pmatrix}$

By applying the formula to calculate the determinant we obtain $\left|B\right|=-18$.

The matrix is invertible, so we can calculate its inverse.
$C^{T}=\begin{pmatrix} 1 & 4 & 1\\ 3 & 1 & 2\\ 2 & 1 & 3 \end{pmatrix}$

We replace every element of the transpose with its cofactor.
$1\longrightarrow (-1)^{1+1}\cdot \Delta_{1,1}=(-1)^{2}\cdot \begin{vmatrix} 1 & 2\\ 1 & 3 \end{vmatrix} = 3 - 2 = 1$

$4\longrightarrow (-1)^{1+2}\cdot \Delta_{1,2}=(-1)^{3}\cdot \begin{vmatrix} 3 & 2\\ 2 & 3 \end{vmatrix} = -(9-4)=-5$

$1\longrightarrow (-1)^{1+3}\cdot \Delta_{1,3}=(-1)^{4}\cdot \begin{vmatrix} 3 & 1\\ 2 & 1 \end{vmatrix} = 3-2=1$

$3\longrightarrow (-1)^{2+1}\cdot \Delta_{2,1}=(-1)^{3}\cdot \begin{vmatrix} 4 & 1\\ 1 & 3\\ \end{vmatrix} = -(12-1)=-11$

$1\longrightarrow (-1)^{2+2}\cdot \Delta_{2,2}=$ $(-1)^{4}\cdot\begin{vmatrix} 1 & 1\\ 2 & 3\\ \end{vmatrix}=3-2=1$

$2\longrightarrow (-1)^{1+3}\cdot \Delta_{2,3}=$ $(-1)^{5}\cdot\begin{vmatrix} 1 & 4\\ 2 & 1 \end{vmatrix}= -(1-8)=7$

$2\longrightarrow (-1)^{3+1}\cdot \Delta_{3,1}=$ $(-1)^{4}\cdot\begin{vmatrix} 4 & 1\\ 1 & 2 \end{vmatrix}=8-1=7$

$1\longrightarrow (-1)^{3+2}\cdot \Delta_{3,2}=$ $(-1)^{5}\cdot \begin{vmatrix} 1 & 1\\ 3 & 2 \end{vmatrix}=-(2-3)=1$

$3\longrightarrow (-1)^{3+3}\cdot \Delta_{3,3}=$ $(-1)^{6}\cdot\begin{vmatrix} 1 & 4\\ 3 & 1 \end{vmatrix}=1-12=-11$

$adj(A)= \begin{pmatrix} 1 & -5 & 1\\ -11 & 1 & 7\\ 7 & 1 & -11 \end{pmatrix}$

$A^{-1} = - \frac{1}{18}\cdot \begin{pmatrix} 1 & -5 & 1\\ -11 & 1 & 7\\ 7 & 1 & -11 \end{pmatrix} =$ $\begin{pmatrix} - \frac{1}{18} & \frac{5}{18} & -\frac{1}{18}\\ \frac{11}{18} & -\frac{1}{18} & -\frac{7}{18}\\ -\frac{7}{18} & -\frac{1}{18} & \frac{11}{18} \end{pmatrix}$

## Properties of the Inverse of a Matrix

If A is an invertible matrix, then:
$A\cdot A^{-1} = A^{-1}\cdot A=I_{n}$

Example 49
$A=\begin{pmatrix} 1 & 3\\ 2 & 5 \end{pmatrix}$

$A^{-1}= \begin{pmatrix} -5 & 3\\ 2 & -1 \end{pmatrix}$

$A\cdot A^{-1}= \begin{pmatrix} 1 & 3\\ 2 & 5 \end{pmatrix} \begin{pmatrix} -5 & 3\\ 2 & -1 \end{pmatrix}=$ $\begin{pmatrix} 1\cdot(-5)+3\cdot2 & 1\cdot3 + 3\cdot(-1)\\ 2\cdot(-5)+5\cdot2 & 2\cdot3 +5\cdot(-1) \end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}= I_{2}$

$A^{-1}\cdot A= \begin{pmatrix} -5 & 3\\ 2 & -1 \end{pmatrix} \begin{pmatrix} 1 & 3\\ 2 & 5 \end{pmatrix}=$ $\begin{pmatrix} -5\cdot1 + 3\cdot2 & -5\cdot3 + 3\cdot 5\\ 2\cdot1 +(-1)\cdot2 & 2\cdot3 +(-1)\cdot5 \end{pmatrix}= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}=I_{2}$

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