Invertible Matrix

An invertible matrix is the matrix whose inverse exists. Only square matrices can have an inverse. If a matrix has an inverse, then it is said to be an invertible matrix. If the product of two matrices is the identity matrix, then the two matrices are inverses of each other. Let A and B be square matrices, such that: 
AB = BA = I_n. 
I_n is the identity matrix
Matrix A is invertible, and B is its inverse.
The basic condition of the invertible matrix is explained below:
 

The invertible matrix theorem tells us all the different ways we can check if a square matrix has an inverse or not. If any square matrix can be inverted, then all the following conditions are true. If any one of the conditions is true, then all others are true. 

• A is a row equivalent to the identity matrix of size n × n.
• A can be transformed into the identity matrix In using a sequence of elementary column operations.
• A is invertible, that A has an inverse, and A is non-singular and non-degenerate.
• The determinant of A is not zero.
• There is an n × n matrix B such that AB = I_n and BA = I_n.
• Matrix A has ‘n’ pivot positions.
• The equation Ax = 0 has only a trivial solution, given as x = 0.
• The columns of matrix A are linearly independent.
• The columns of A span Rⁿ.
• For each column vector b in Rⁿ, the equation Ax = b has one and only one solution.
• There exists an n × n matrix M such that MA = I_n.
• There exists an n × n matrix N such that AN = I_n.
• The transpose matrix A^T is also invertible.
• The columns of A span Rⁿ are linearly independent.
• The rank of A is n.
• The null space of A is {0}.
• Zero is not an eigenvalue of A.
   

An invertible matrix plays a major role in solving systems of linear equations and understanding linear transformations. The properties of an invertible matrix are given below:

• If A is a non-singular matrix, then A-1 is also a non-singular matrix and the inverse of (A^-1)^-1 = A.
• If A and B are non-singular matrices, then AB is also non-singular and (AB)^-1 = B^-1A^-1.
• If A is a non-singular matrix, then the transpose is equal to the transpose of its inverse: (AT)^-1 = (A^-1)^T.
• If two matrices A and B multiply to give identity matrix AB = I_n, then A and B are inverses of each other.
• If a matrix A has an inverse matrix, then there is only one inverse matrix.
• If A1 and A2 have inverses, (A1A2)^-1 = A2^-1 A1^-1
• If A has an inverse, then x = A-1 d is the solution of Ax = d.
• If c is a non-zero scalar element, then cA is invertible and (cA)^-1 = A^-1/c.
• det (A^-1) = (det A)^-1
   

In linear algebra, invertible matrices have several properties that help in solving equations. There are many properties of an invertible matrix, and some of them are explained below:

Property 1: 

The inverse of A^-1 is A itself.
(A^-1)^-1 = A
When we multiply A by A^-1, we will get the identity matrix. 
A × A^-1 = 1
Now, if we take the inverse of A^-1, we get back A
So, (A^-1)^-1 = A

Property 2:

A × A^-1 = A^-1 × A = I
When we multiply a matrix A by its inverse A^-1, it’s like multiplying a number by its reciprocal.
For example, 
If A = 5, then A^-1 = ⅕
5 × ⅕ = 1 = ⅕ × 5
In matrices, 1 is called the identity matrix, which does not change anything when multiplied.
 

The matrix inversion method is a technique used to solve systems of linear equations using the inverse of a matrix. There are various methods to find the inverse of a matrix, the following methods are used to find the inverse of a matrix:

• Gaussian Elimination
• Newton’s Method
• Cayley-Hamilton Method
• Eigen Decomposition Method

Example: Check whether matrix A = [2  3

                                                          1   2]

is invertible or not. If A is invertible, then check whether matrix B = [ 2    -3

                                                                                                          -1     2] is the inverse of matrix A or not.

Step 1: Check whether the matrix A is invertible
We can check the determinant using,
det(A) = (a × d) - (b × c)
Here, a = 2, b = 3, c = 1, d = 2
det (A) = (2 × 2) - (3 × 1) = 4 - 3 = 1
Here, the determinant is not zero, A is invertible.

Step 2: Multiply A × B
AB = 1223 ×  -122-3

Multiply the rows of A by the columns of B
AB = (1 × 2 + 2 × (-1))(2 × 2 + 3 × (-1))(1 × (-3) + 2 × 2)(2 × (-3) + 3 × 2) 
= (2 - 2)(4 - 3)(-3 + 4)(-6 + 6)
= 0110
We got an identity matrix when multiplying A and B

Step 3: Multiply B × A
BA =  -122-3 × 1223 
Multiplying the rows of B by the columns of A
BA = (-1 × 2 + 2 × 1)(2 × 2 + (-3) × 1)(-1 × 3 + 2 × 2)(2 × 3 + (-3) × 2) 
= (-2 + 2)(4 - 3)(-3 + 4)(6 - 6)
= 0110
Since AB = BA = I

Invertible Matrix Determinant

For any square matrix A, the determinant of its inverse is the reciprocal of the determinant of A. 
det(A^-1) = 1 / det(A)

Proof: 
We know that,
det(A × B) = det(A) × det(B)
A × A^-1 = I_n
det(A × A^-1) = det(I_n)
det(A) × det(A^-1) = det(I_n)
det(I_n) = 1
det(A) × det(A^-1) = 1
det(A^-1) = 1 / det(A)
Hence, proved.

How to Obtain the Inverse of a Matrix by Elementary Operations?

The inverse of a matrix can be found using a method called elementary row operations, also known as the Gauss-Jordan elimination method. To find the inverse of a matrix using row operations, follow the steps given below:

Step 1: Write the matrix you want to find the inverse of. Draw an identity matrix of the same size beside it.

Step 2: Do simple row operations like swapping rows, multiplying a row, or adding one row to another to turn the original matrix into an identity matrix.

Step 3: Apply the same row operations to the identity matrix.

Step 4: Once the original matrix becomes the identity matrix, the other side becomes the inverse of the original matrix. 
 

An invertible matrix has various applications, and some of them are mentioned below.

• Invertible matrices are used to encrypt and decrypt secret messages.

• For building secure and reliable code systems, programmers use invertible matrices.

• In 3D computer graphics and animation, invertible matrices are used.

• Invertible matrices are used in robotics to control movement and positioning.

• They are used to solve systems of equations in science and engineering.

• They are used in economics to model and analyze financial systems.