The 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 significant 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−1), the resulting product is the Identity matrix (denoted by I). To illustrate this concept, see the diagram below.
- 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 I. That means invertible matrices are commutative.
Formula to Find the Inverse of a 2×2 Matrix
Given matrix A…
Its inverse is calculated using the formula…
where: det A is the determinant of matrix A
Few observations about the formula:
- Entries “a” and “d” from matrix A are swapped in terms of location in the formula.
- Entries “b” and “c” from matrix A remain in their current positions, however, reversed in signs in the formula. In other words, put negative symbols in front of entries “b” and “c”.
- Since det A is just a number, then is also a number that would serve as the scalar multiplier to the matrix in the formula. See my separate lesson how to multiply a scalar into a matrix.
Examples of How to Find the Inverse of a 2×2 Matrix
Example 1: Find or solve the inverse of matrix , 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 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 det A = −1 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 (separate tutorial) of A and A−1 two ways, and see if we’re getting the Identity matrix.
Since the multiplication both ways generates the Identity matrix, then we are guaranteed that the inverse matrix obtained using the formula is the correct answer!
Example 2: Find or solve the inverse of matrix , if it exists.
First, find the determinant of matrix B.
Second, 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 or solve the inverse of matrix , if it exists.
This is a great example where the determinant is neither +1 nor −1 which would usually result 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.
Here we go…
Step 1: Find the determinant of matrix C.
- The formula to find the determinant
- Animated solution to find the determinant of matrix C
Step 2: The determinant of matrix C equals −2. Plug into the formula, and simplify to get the inverse of matrix C.
Step 3: Check if the solved 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.
Example 4: Find or solve the inverse of matrix , if it exists.
In our previous three examples, we are successful in finding the inverse of the given 2×2 matrices. I don’t want to give the impression that all 2×2 matrices have inverses.
In this example, I want to illustrate when a given 2×2 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 or solve the inverse of matrix , 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.
Step 3: Verify your answer by checking that you get the Identity matrix in both cases: