Lesson Objectives
• Learn how to find the inverse of a matrix

## How to Find the Inverse of a Matrix

In this lesson, we will learn how to find the inverse of a matrix. First and foremost, only a square matrix or a matrix with the same number of rows as columns can have an inverse. When a matrix has an inverse, it is known as an invertible or a nonsingular matrix. When a matrix does not have an inverse, it is known as a singular matrix.

### Identity Matrix

When we have a square matrix whose diagonal entries are 1's while all other entries are zeros, we refer to this as an identity matrix. Generally, this is notated with In, where n is the size or order of the matrix. Since these identity matrices are square, we only need one number.

#### 2 x 2 Identity Matrix:

$$I_{2}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array}\right]$$

#### 3x 3 Identity Matrix:

$$I_{3}=\left[ \begin{array}{ccc}1&0 & 0\\ 0&1 & 0 \\0 & 0 & 1\end{array}\right]$$ When we multiply a square matrix by an identity matrix of the same order, this will leave the matrix unchanged. $$A=\left[ \begin{array}{cc}3&-5\\ 6&5\end{array}\right]$$ $$I_{2}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array}\right]$$ $$AI_{2}=\left[ \begin{array}{cc}3&-5\\ 6&5\end{array}\right]$$

### Finding the Inverse of a Matrix

When we multiply a matrix by its inverse, we get the identity matrix. $$AA^{-1}=I_{n}$$ To find the inverse of a nonsingular matrix A:
• Form the augmented matrix: [A | In]
• Use row operations to obtain: [In | B]
• B is the inverse of A
To find the inverse for a 2 x 2 matrix, we have a shortcut method: $$A=\left[ \begin{array}{cc}a&b\\ c&d\end{array}\right]$$ $$A^{-1}=\frac{1}{ad - bc}\left[ \begin{array}{cc}d&-b \\ -c&a\end{array}\right]$$ Let's look at an example.
Example #1: Find the inverse. $$A=\left[ \begin{array}{cc}7&-3\\ 4&-1\end{array}\right]$$ $$A^{-1}=\left[ \begin{array}{cc}-\frac{1}{5}&\frac{3}{5}\\ -\frac{4}{5}&\frac{7}{5}\end{array}\right]$$

#### Skills Check:

Example #1

Find the inverse of A. $$A=\left[ \begin{array}{cc}-6&-2\\ -1 &1\end{array}\right]$$

A
$$A^{-1}=\left[ \begin{array}{cc}-\frac{1}{8}&-\frac{1}{4}\\ -\frac{1}{8}&\frac{3}{4}\end{array}\right]$$
B
$$A^{-1}=\left[ \begin{array}{cc}-\frac{1}{3}&\frac{3}{7}\\ -\frac{2}{9}&\frac{7}{5}\end{array}\right]$$
C
$$A^{-1}=\left[ \begin{array}{cc}-\frac{7}{5}&\frac{9}{5}\\ -\frac{2}{5}&\frac{7}{5}\end{array}\right]$$
D
$$A^{-1}=\left[ \begin{array}{cc}6&4 \\ -\frac{4}{5}&\frac{3}{5}\end{array}\right]$$
E
$$A^{-1}=\left[ \begin{array}{cc}-\frac{2}{5}&\frac{7}{5}\\ 9&13\end{array}\right]$$

Example #2

Find the inverse of A. $$A=\left[ \begin{array}{cc}10&-4\\ 2&-6\end{array}\right]$$

A
$$A^{-1}=\left[ \begin{array}{cc}-\frac{1}{9}&11\\ -9&-8\end{array}\right]$$
B
$$A^{-1}=\left[ \begin{array}{cc}-\frac{2}{7}&\frac{1}{9}\\ \frac{2}{11}&\frac{13}{5}\end{array}\right]$$
C
$$A^{-1}=\left[ \begin{array}{cc}-\frac{12}{5}&\frac{7}{5}\\ -\frac{9}{5}&\frac{1}{5}\end{array}\right]$$
D
$$A^{-1}=\left[ \begin{array}{cc}\frac{3}{26}&-\frac{1}{13}\\ \frac{1}{26}& -\frac{5}{26}\end{array}\right]$$
E
$$A^{-1}=\left[ \begin{array}{cc}8&11\\ -\frac{3}{5}&\frac{9}{5}\end{array}\right]$$