Invertible Matrices

What is Invertible Matrix

An array of numbers arranged in the form of rows and columns is known as the matrix. Dimensions of a matrix are given by the number of rows and columns of a matrix, given as m x n where m and n represent the number of rows and columns respectively. The basic mathematical operations like addition, subtraction, multiplication, and division can be done on matrices. There are different types of matrices and have different applications. Here, we will study what is invertible matrix or the invertible matrices.

An invertible matrix is also known as a non-singular or non-degenerate matrix. An invertible matrix cannot have its determinant value as 0.

Definition of Invertible Matrix

A matrix 'A' of dimension n x n is called invertible only under the condition, if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order. In such a case matrix B is known as the inverse of matrix A. Inverse of matrix A is symbolically represented by 'A^-1'. 

For example, if matrix A and B satisfy this condition AB=BA=I, then we can say B is the inverse of A written as A^-1=B. Similarly, we can also say A is the inverse of B written as B^-1

How to Determine if a Matrix is Invertible

An n x n square matrix M is not invertible precisely if det M is 0 which is the determinant value of M is 0, which occurs precisely if the rows (or columns) are not linearly independent, which in turn occurs precisely if the rank of M is not n.

A matrix that has no inverse is singular. When the determinant value of square matrix I exactly zero the matrix is singular.

Invertible Matrix Theorem

Theorem1: Unique inverse is possessed by every invertible matrix.

Proof: Let there be a matrix A of order n×n which is invertible. Let two inverses of A be B and C

Then,AB=BA=In..(1) (In=identity matrix of order n)

and AC=CA=In….(2)

Now,AB=In

=> C(AB)=CIn (premultiplying by C)

=> (CA)B=CIn (by associativity)

=> InB=CIn (since CA=In from (2))

=> B=C.      (since InB=B,CIn=C)

So, this proves that an invertible matrix possesses a unique inverse.

Theorem2: A square matrix is invertible if and only if it is non-singular.

Proof : Suppose there Is an invertble matrix A.Then there exists a matrix B such that

     AB=In=BA=>|A|=|In|

=> |A| |B|=1    ( since |AB|=|A||B|)

=> |A| is not equal to 0 => A is non-singular matrix.

Conversely this can also be proved.

Theorem3: If A and B are invertible matrices and of the same order and are, then (AB)^-1 = B^-1A^-1.

Proof:

(AB)(AB)^-1 = I                                     (Taken from the inverse matrix definition)

A^-1 (AB)(AB)^-1 = A^-1 I                    (Multiplying A^-1 on both sides)

(A^-1 A) B (AB)^-1 = A^-1                      (A^-1I=A)

I B (AB)^-1 = A^-1

B (AB)^-1 = A^-1

B^-1 B (AB)-1 = B^-1 A^-1

I (AB)^-1 = B^-1 A^-1

(AB)^-1 = B^-1 A^-1

Invertible Matrix Example

1)Prove that B is inverse of matrix A

A = \[\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}\] B = \[\begin{bmatrix} 5 & -2 \\ -2 & 1 \end{bmatrix}\]  

Solution=>  On multiplying A with B we get an identity matrix

AB = \[\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}\] \[\begin{bmatrix} 5 & -2 \\ -2 & 1 \end{bmatrix}\] = \[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]

 Similarily when we multiply B with we get an identity matrix

BA = \[\begin{bmatrix} 5 & -2 \\ -2 & 1 \end{bmatrix}\] \[\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}\] = \[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]

Hence we can conclude from above that AB=BA=I

so we can say that B is inverse of A i.e. A^-1=B

Also, we can say that A is inverse of B i.e.B^-1=A

2) If A is a non-singular matrix, then prove that 

|A^-1|=|A|^-1 i.e. |A^-1|= 1

                                       |A|

Solution=> Since |A| is not equal to zero,therefore A^-1 exists such that

AA^-1=I =A^-1A => |AA^-1|=|I|

=> |A| |A^-1|=1 (since|AB|=|A||B| and |I|=1)

=> |A¹|= 1

               |A|     (since |A| is not equal to zero)

 Matrix Inversion Methods

These are the methods by which we find other matrix B which is the inverse of matrix A and satisfies invertible matrix equation AB=BA=I

The methods are

1. Gaussian Elimination

2. Newton's Method

3. Cayley Hamilton Method

4. Eigen Decomposition Method

Applications of Invertible Matrices

The practical applications, where the solution for the system of the equation should be unique there it is necessary that the matrix involved should be invertible. Such applications are:

a) Least-squares or Regression

b) Simulations

c) MIMO Wireless Communications