# Inverse of a 2×2 Matrix

In this lesson, we are only going to deal with **2×2 square matrices**. I have prepared five (5) worked examples to illustrate the procedure on how to solve or find the inverse matrix using the **Formula Method**.

Just to provide you with the general idea, two matrices are inverses of each other if their product is the **identity matrix**. An identity matrix with a dimension of 2×2 is a matrix with zeros everywhere but with 1’s in the diagonal. It looks like this.

It is important to know how a matrix and its inverse are related by the result of their product. So then,

- If a 2×2 matrix A is invertible and is multiplied by its inverse (denoted by the symbol
**A**), the resulting product is the Identity matrix which is denoted by [latex]I[/latex]. To illustrate this concept, see the diagram below.^{−1}

- In fact, I can switch the order or direction of multiplication between matrices A and A
^{−1}, and I would still get the Identity matrix [latex]I[/latex]. That means invertible matrices are commutative.

How do we find the inverse of a matrix? The formula is rather simple. As long as you follow it, there shouldn’t be any problem. Here we go.

## The Formula to Find the Inverse of a 2×2 Matrix

Given the matrix A

Its inverse is calculated using the formula

where [latex]\color{red}{\rm{det }}\,A[/latex] is read as the determinant of matrix A.

A few observations about the formula:

- Entries [latex]\color{blue}a[/latex] and [latex]\color{blue}d[/latex] from matrix A are swapped or interchanged in terms of position in the formula.
- Entries [latex]\color{blue}b[/latex] and [latex]\color{blue}c[/latex] from matrix A remain in their current positions, however, the signs are reversed. In other words, put negative symbols in front of entries [latex]b[/latex] and [latex]c[/latex].
- Since [latex]\color{red}{\rm{det }}\,A[/latex] is just a number,
then [latex]\large{1 \over {{\rm{det }}A}}[/latex]is also a number that would serve as the scalar multiplier to thematrix

See my separate lesson on scalar multiplication of matrices.

### Examples of How to Find the Inverse of a 2×2 Matrix

**Example 1:** Find the inverse of the 2×2 matrix below, if it exists.

The formula requires us to find the determinant of the given matrix. Do you remember how to do that? If not, that’s okay. Review the formula below on how to solve for the determinant of a 2×2 matrix.

So then, the determinant of matrix A is

To find the inverse, I just need to substitute the value of [latex]{\rm{det }}A = – 1[/latex] into the formula and perform some “reorganization” of the entries, and finally, perform scalar multiplication.

- Here goes again the formula to find the inverse of a 2×2 matrix.

- Now, let’s find the inverse of matrix A.

Let’s then check if our inverse matrix is correct by performing matrix multiplication of **A** and **A ^{−1}** in two ways, and see if we’re getting the Identity matrix.

Since multiplying both ways generate the Identity matrix, then we are guaranteed that the inverse matrix obtained using the formula is the correct answer!

**Example 2:** Find the inverse of the 2×2 matrix below, if it exists.

First, find the determinant of matrix B.

Secondly, substitute the value of **det B = 1** into the formula, and then reorganize the entries of matrix B to conform with the formula.

I will leave it to you to verify that

In other words, the matrix product of **B** and **B ^{−1}** in either direction yields the Identity matrix.

**Example 3:** Find the inverse of the matrix below, if it exists.

This is a great example because the determinant is neither [latex]+1[/latex] nor [latex]−1[/latex] which usually results in an inverse matrix having rational or fractional entries. I must admit that the majority of problems given by teachers to students about the inverse of a 2×2 matrix is similar to this.

**Step 1**: Find the determinant of matrix C.

- The formula to find the determinant

- Below is the animated solution to calculate the determinant of matrix C

**Step 2**: The determinant of matrix C is equal to [latex]−2[/latex]. Plug the value in the formula then simplify to get the inverse of matrix C.

**Step 3**: Check if the computed inverse matrix is correct by performing left and right matrix multiplication to get the Identity matrix.

Yep, matrix multiplication works in both cases as shown below.

First case:

Second case:

**Example 4:** Find the inverse of the matrix below, if it exists.

In our previous three examples, we were successful in finding the inverse of the given [latex]2 \times 2[/latex] matrices. I don’t want to give you the impression that all [latex]2 \times 2[/latex] matrices have inverses.

In this example, I want to illustrate when a given [latex]2 \times 2[/latex] matrix fails to have an inverse. How does that happen?

If we review the formula again, it is obvious that this situation can occur when the determinant of the given matrix is zero because **1 divided by zero is undefined**. And so, an undefined term distributed into each entry of the matrix does not make any sense.

Let’s go back to the problem to find the determinant of matrix D.

Therefore, **the inverse of matrix D does not exist** because the determinant of D equals zero. This is our final answer!

**Example 5:** Find the inverse of the matrix below, if it exists.

**Step 1**: Find the determinant of matrix E.

**Step 2**: Reorganize the entries of matrix E to conform with the formula, and substitute the solved value of the determinant of matrix E. Distribute the value of [latex]\large{1 \over {{\rm{det }}E}}[/latex] to the entries of matrix E then simplify, if possible.

**Step 3**: Verify your answer by checking that you get the Identity matrix in both scenarios.

First scenario:

Second scenario:

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