Introduction

An invertible matrix is also known as a non-degenerate and non-singular matrix. In our Invertible matrices article, we will discuss the definition of invertible matrices, their properties, and theorems, along with a step-by-step guide on calculating the inverse of a 2×2 matrix and 3×3 matrix. We will also see how the inverse of a matrix can be calculated by elementary comparison. 

Definition of invertible matrices

• A matrix is a rectangular array of numbers that are arranged in rows and columns.
• Let there be a matrix A of order n × n and another matrix B of the same dimension. Matrix A will be considered an invertible matrix if AB = BA = I_n, where n denotes the order of the matrix. In this scenario, matrix B is the inverse of matrix A.
• The inverse of matrix A, which is an invertible matrix, can be represented as A^-1.
• An invertible matrix is also known as a non-degenerate matrix or a non-singular matrix.
• Similarly, a square matrix that is non-invertible is referred to as a degenerate or singular matrix.

Inverse of a 2×2 matrix

• Let matrix A =

• The formula to calculate the inverse of matrix A = A^-1 =  

• We can observe in the above formula that entries p and s swapped their positions with each other. We can also observe that the signs of q and r were reversed.
• Let’s see this formula in action.

Find the inverse of matrix A =

Inverse of matrix A =

Therefore, Det (A) =  =  72 – 80 = -8

Inverse of matrix A =    

Inverse of 3×3 matrix

Finding the inverse of the 3×3 matrix is not as easy as finding the inverse of a 2×2 matrix and involves several steps. Let us solve the following matrix to understand the concept of inverting a 3×3 matrix.

A=

• Step 1 is to find the determinant of A.

Det (A) = 1 (0-24) – 2(0-20) + 3(0-5)

Det (A) = -24 + 40 – 15 = 1

Since det (A)  0. Therefore, matrix A is invertible.

• Transpose of the original matrix =

• Step 2 is to calculate the determinant of every 2×2 matrix within the original matrix.

Determinant for 2×2 matrix for all of the first-row elements-

A₁₁=  = -24

A₁₂ =  = -18

A₁₃= = 5

Second row-

A₂₁= = -20

A₂₂= =-15

A₂₃= =4

Third row-

A₃₁=  = -5

A₃₂= = -4

A₃₃= =1

• The next step is to form a matrix from the cofactors. Therefore, we get the new matrix
• Adjoint matrix = × =
• Inverse of matrix A = A^-1 = ,

Therefore, A^-1 =

• Therefore, the inverse of matrix A =

Theorems of invertible matrices

• The inverse of an invertible matrix is always unique.
• If A and B are two separate invertible matrices of the same order, then (AB)^-1 = B^-1 A^-1.
• Matrix A is an invertible matrix only when it’s determinant does not equal zero.
• The transpose of a matrix is also an invertible matrix.

Properties of invertible matrices

• If A is an invertible matrix, then its inverse is also invertible.
• The inverse of an inverse matrix is the original matrix. For example (B^-1)^-1 = B.
• If A and B are two invertible matrices, then their product is also invertible. The inverse of AB can be written as- (AB)^-1 = B^-1 A^-1.
• If A and B are matrices of the same order and I_n = AB, then A and B are inverses of each other.

How to obtain the inverse of a matrix by elementary operations?

• The inverse of a matrix by elementary operations or the Gaussian elimination method, or the row reduction method, is an algorithm used to solve a system of linear equations.
• Elementary operations or the row reduction method is used to find the inverse of a matrix, the determinant of a matrix, and the rank of a matrix.
• The steps involved in finding the inverse of a matrix using elementary operations are-
• Step 1- Create an identity matrix of the same order as the original matrix. Therefore, if the original matrix is 2×2, then the identity matrix drawn will also be 2×2.
• Step 2- In this step, you will make the original matrix equivalent to the identity matrix by performing row or column operations.
• Step 3- Perform either row or column operations on the identity matrix. The resultant identity matrix is the inverse of the original matrix.
• Let us find the inverse of the following matrix by performing elementary operations.

A =,

a₁₁= 2, a₁₂= 0, a₁₃= 3,  a₂₁= -1, a₂₂= 3, a₂₃= -4, a₃₁= -3, a₃₂= 1, and a₃₃= -4.

We start by interchanging Rows 2 and 3 so that a₂₂= 1.

Therefore,

Then we substitute the value of R₁ with R₁ + R₃, so that a₁₁= 1

Then we substitute the value of R2 with R₂ – 3R₃ so that a₂₁= 0

The next step is to substitute the value of R3 with R₃ + R₁, so that a₃₁= 0

The next step will be to make the value of a₂₂= 0, and for that, we will divide the second row with -8.

To make the value of a₁₃= 0, we will subtract R2 from R1.

In the next step, we will substitute the value of R₃ with R₃ – 3R₂, this will make the value a₃₂= 0 and a₃₃= 1.

We will add row 2 with row 3, to make a₂₃= 0

And the last step is to make a₁₂= 0 by R₁ – 2R₂

After transforming the original matrix into an identity matrix, we will perform all of the same operations on the identity matrix.

After performing all of the same operations on the identity matrix, we are able to determine the inverse of A.

A^-1 =

You should try this exercise to check if your answer matches ours after performing all of the same operations on the identity matrix.

Conclusion

With this, we conclude our article on Invertible Matrices. An invertible matrix is also known as a non-degenerate and non-singular matrix. A matrix is a rectangular array of numbers that are arranged in rows and columns.

Let there be a matrix A of order n × n and another matrix B of the same dimension. Matrix A will be considered an invertible matrix if AB = BA = In, where n denotes the order of the matrix. In this scenario, matrix B is the inverse of matrix A. The inverse of matrix A, which is an invertible matrix, can be represented as A^-1. An invertible matrix is also known as a non-degenerate matrix or a non-singular matrix. Similarly, a square matrix that is non-invertible is referred to as a degenerate or singular matrix.