What Is a Matrix Inverse?

From the word inverse itself, we can infer that it means undo an action. For example if you multiply $a$ to $b$, then to "undo" this you want to multiply $\frac{1}{a}$ to the result: $\frac{1}{a}\cdot a\cdot b = (\frac{1}{a}\cdot a)\cdot b = b$. Similarly, when we want to find the inverse of matrix, we want to inverse of the matrix cancel out the effect of the matrix. I assume you already know what is an identity matrix, which has similar position as 1 to numbers, then the definition of inverse of a matrix will seem to be natural: $$\begin{align} \text{ Suppose A and B are two $n\times n$ matrices, }\\ \text{ $A$ is the inverse of $B$ if and only if $ AB = BA = I_{n}$}\\ \text{We denote the inverse of matrix $A$ as $A^{-1}$}\end{align}$$

(Video) Inverse of a $3\times3$ Matrix

by Math Meeting

Here's a video by Math Meeting showing you how to find the inverse of a $3\times3$ matrix using the determinant. It's a lot more complicated than a $2\times2$ matrix.

(Video) Inverse Matrix Using Gauss-Jordan and Row Reduction

by PatrickJMT

Here are some videos by patrickJMT showing you a different method of finding the inverse of a matrix. Note that for a $2\times2$ matrix it's generally easier to use the determinant method. But for a $3\times3$ or higher matrix, I find it simpler to use this method. Both will work so use whatever one you can do the quickest or easiest.

(Video) Inverse Matrices, Column Space, and Null Space

by 3Blue1Brown

What is an inverse?

While formulas and calculations are important, so is intuition. Below is a video that presents the intuition on matrix inverses, column space, and null space, though not how to calculate them.

Note: when reading the formulas below, please remember that $A^{-1}$ means the inverse matrix of $A$.

For a 2x2 matrix:

For a 2x2 matrix $$\begin{bmatrix} a&b\\c&d\end{bmatrix}$$ You can do $$A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d&-b\\-c&a\end{bmatrix}$$ So, for example, to find the inverse of the matrix $$\begin{bmatrix}-1&-3\\4&5\end{bmatrix}$$ you would do $$A^{-1} = \frac{1}{(-1*5)-(3*-4)}\begin{bmatrix}5&3\\-4&-1\end{bmatrix}$$ which then reduces to $$A^{-1} = \frac{1}{7}\begin{bmatrix}5&3\\-4&-1\end{bmatrix}$$ and then $$A^{-1} = \begin{bmatrix}\frac{5}{7}&\frac{3}{7}\\ \frac{-4}{7}& \frac{-1}{7}\end{bmatrix}$$ which is the inverse of your original matrix.

I will soon update this to include 3x3 matrices and how to find their inverse.

Inverse

In general, the mathematical definition of an inverse is a reciprocal of a value. For example, the inverse (or reciprocal) of $4$ is $\frac14$. We can denote the inverse of a value by raising the value to the power of $-1$. So, $4^{-1} = \frac14$. What's cool about inverses is that if you multiply the original value by the inverse, you get a product of $1$. $4 \cdot \frac14 = 1$. This is given by the Identity Property of Multiplication.

We have also seen inverses on graphs, where we reflect a given function over the line $y=x$. We can also find the inverse algebraically.

Similarly, we can calculate inverses of matrices. Similar to numerical values, we denote the inverse of a given matrix $A$ with $A^{-1}$.

Matrix Inverse

Looking more specifically at matrix inverses, there are two key notes:

• If you multiply a given matrix $A$ with its inverse $A^{-1}$, you will get a product of the identity matrix. So: $AA^{-1}=I$

• To find the inverse, we use the following formula:

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

You'll notice that $\frac{1}{ad-bc}$ is the determinant!

Example

Let's test these notes out with an example matrix.

Given matrix $K$,

$$\begin{bmatrix} 1&3 \\ 2&5 \\ \end{bmatrix}$$

We can find the inverse by using our formula.

$$\begin{bmatrix} 1&3 \\ 2&5 \\ \end{bmatrix} = \frac{1}{(1)(5)-(3)(2)} \begin{bmatrix} 5&-3 \\ -2&1 \\ \end{bmatrix}$$

We'll first simplify the determinant:

$$= \frac{1}{5-6} \begin{bmatrix} 5&-3 \\ -2&1 \\ \end{bmatrix}$$

Which can be further simplified to:

$$= -1 \begin{bmatrix} 5&-3 \\ -2&1 \\ \end{bmatrix}$$

Always watch your signs! Next, we will use scalar multiplication to simplify this further:

$$\begin{bmatrix} -5&3 \\ 2&-1 \\ \end{bmatrix}$$

We've found our inverse matrix, $K^{-1}$! $K^{-1} = \begin{bmatrix} -5&3 \\ 2&-1 \\ \end{bmatrix}$

To check our work, we can find multiply the two matrices together ,$KK^{-1}$, and if the product is an identity matrix, we know our work is correct.

$$\begin{bmatrix} 1&3 \\ 2&5 \\ \end{bmatrix} \begin{bmatrix} -5&3 \\ 2&-1 \\ \end{bmatrix} = \begin{bmatrix} 1&0 \\ 0&1 \\ \end{bmatrix}$$

Our product is a $2x2$ identity matrix!

Matrix Inverse

Before finding a matrix inverse, it’s important to understand basic matrix operations like adding and subtracting matrices. Also make sure you have reviewed zero matrices.

To calculate the inverse of a matrix, we have a special formula that we use! You may see this method when solving systems of equations.

It may be helpful to review the determinant of the matrix.

Example

Find the inverse of matrix $B$.

$$B = \begin{bmatrix} 1&2 \\ 5&6 \\ \end{bmatrix}$$

Let's plug in our values into the formula!

$$B^{-1} = \frac{1}{1*6 - 2*5} \begin{bmatrix} 6&-2 \\ -5&1 \\ \end{bmatrix}$$

We can simplify,

$$B^{-1} = \frac{1}{-4} \begin{bmatrix} 6&-2 \\ -5&1 \\ \end{bmatrix}$$

You may be wondering why we need to know how to find the inverse. The invertible matrix theorem explains when a matrix inverse exists.