Last Updated on October 21, 2026 by Maged kamel
The Inverse of a Matrix, the Inverse of a 2×2 Matrix.
I have added a video to illustrate the content of this Post.
The video timestamps are shown.
- 00:00:00- Introduction to the Inverse of a 2×2 matrix.
- 00:00:35 – How can we use an augmented Matrix to find the Matrix Inverse?
- 00:02:35 – What are the elementary Row operations?
- 00:03:45 – Solve a practice Problem for the Inverse of a Matrix using an augmented Matrix
- 00:08:58 – Find the Inverse of a Matrix using the determinant and the cofactor.
- 00:10:22- The Types of Elementary Matrices.
- 00:16:08- Apply Elementary matrices to find the Inverse of a Matrix.
A brief description of the content of the lecture.
The objective of this lecture is to determine the Inverse of a Matrix using elementary Row operations and use it to solve simultaneous equations by lower/upper factorization or Doolittle’s factorization. How can we use elementary matrices to get the Inverse of a 2×2 Matrix?
How to find the Inverse of a Matrix?
First, let us discuss how to compute the Inverse of a Matrix using elementary Row operations. We use the Augmented Matrix on the left-hand side, written as A | I.
For example, it can be 3 by 3, 2 by 2, or 4 by 4, and it’s written with a Capital L.
Using elementary operations, we convert the original Matrix A to the identity Matrix I, and the same operations convert the I in the augmented Matrix on the right-hand side into A-1, the Inverse of A.
Our original Matrix, which is the one we are looking for, is the set of elementary Row operations.
The possible operations on a Matrix.
There are three operations: swapping rows (e.g., R1 and R2), multiplying or dividing each element in a Row by a constant, and replacing a Row with the sum of that Row and a multiple of another Row.
The 3rd operation is replacing a Row by adding or subtracting multiples of other rows. We can say R1 +2*R2 or R1-R3, where R1 is the first Row, and R3 is the third Row, etc.
A solved example of how to get the Inverse of a 2×2 Matrix.
For example, if we have a 2×2 Matrix A = (-3, -2; 7, 4), we need to find the Inverse of A.
We will use the augmented Matrix method; we write Matrix A on the left-hand side.
We divide the Matrix with a vertical line and place the identity Matrix on the right. In our case, it will be 2 by 2. As we can see, the diagonal is (1, 1), and the other elements are zeros.
We want to convert the left-hand side into (1 1 0 0). How can this be done?
First, to convert -3 to +1, we multiply R1*1/3, and we are going to put the result in place of R1, so (-3*-1/3) becomes 1,(-2*-1/3) becomes +2/3, and the right-hand side will be (-1/3,0).
The second Row remains unchanged because there is no operation, so it keeps the same elements: (7 4 0 1).
The next step is to make this 7 = 0, which we can do by multiplying the first Row by (-7) and adding it to the second Row (R2), so (-7*1 + 7) becomes 0. Here 2/ 3 -7 +4=-2/3, -7*-1/3 will be 7/3, and the application of 0 will be 0 + R2.
The next step is to multiply the second Row by (-3/2) and place it in our Table to zero, which is the same. This is reciprocal, which becomes 1 and 7/3. RI. (-3/2) becomes (-7/3)- 3/2, or the last item, which is equal to 1.
The next target is to make 2/3 = 0; we can do this by multiplying the second Row (which is 1) by -2/3 and adding it to Row 1.
We get (1 0 2 +1), and since the second Row is unchanged, it remains (0 1 – 1/2 (-3/2). On the left-hand side, the diagonal is 1, and the other diagonal is 0.
This is the Inverse we are looking for; it is divided and lies in the right-hand portion of this line.
Another way to find the Inverse of a Matrix is to use the determinant.
There is a rule for the Matrix A: if it is (2×2), we replace it with this: we swap these; -3 goes down, we change the sign of 7 to -7, and -2 becomes +2.
But again, we are going to divide by the determinant, which is(-3*4-1*-2), which at the end will become +2, so the Inverse of the Matrix is called the adjoint divided by the determinant; the adjoint is (4 -2, and (-7 -3/2). The details of the operations are shown on the next slide.
The result we obtained using the determinant matches that obtained using the augmented Matrix.
Introduction to Elementary Matrices.
The following slide image explains the definition of an Elementary Matrix. There are three types of these matrices: the first is E1, obtained by swapping rows of an Identity Matrix. The second one is E2, obtained by multiplying a Row of the identity Matrix by a nonzero constant.
The third elementary Matrix, E3, is obtained from the Identity Matrix by adding a multiple of one Row to another Row.
Find the Inverse of a Matrix by elementary matrices.
For the same example we used to compute the Inverse of a Matrix, we will perform the same operations on Matrix A using elementary matrices. The first operation was to multiply the first Row by (-1/3) and place the result in the same Row. This can be done by multiplying Matrix A by (-1/3 0, 0 1), which is type E1.
The second operation was multiplying the first Row by (-7), adding it to the second Row, and placing the result in the second Row. This can be done by multiplying Matrix A by E1, then multiplying by another elementary Matrix: (1 0, -7 1). This Matrix is the E2 Matrix. Please refer to the image on the next slide for illustration.
The third operation was multiplying the second Row by (- 3/2) and placing the result in the second Row. This can be done by using another elementary Matrix E3. The last operation: multiply (-2/3) by the second Row and add the result to the first Row.
Place the result in Row 1 using the E4 elementary Matrix. Please refer to the next slide image for illustration.
To get the Inverse of Matrix A, we will use the four types of elementary matrices. Multiplying E4*E3*E2*E1 gives the Inverse Matrix. To check that we have the correct solution, we multiply by A and obtain the identity Matrix.
If we multiply the inverses of E1, E2, E3, and E4 by the identity Matrix, we get A. Thanks a lot.
This is the PDF used to illustrate Post 1. You can download and review it.
The following Post is an Easy Illustration of the LU decomposition method for a 2×2 Matrix.
HELM-Helping Engineers Learn Mathematics.
This is the Omni calculator for estimating various linear algebra items: LU Decomposition Calculator.
This is a Link to another Matrix calculator.