Invertible Matrices

In linear algebra, an $n\times n$ square matrix is called invertible, if the product of the matrix and its inverse is the identity matrix.

Definition

A matrix $A$ of dimension $n\times n$ is called invertible if and only 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. Matrix $B$ is known as the inverse of matrix $A$. Inverse of matrix $A$ is symbolically represented by$A^{-1}$.Invertible matrix is also known as a non-singular matrix or non-degenerate matrix.

For example, matrices $A$ and $B$ are given below:

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

Now we multiply $A$ with $B$ and obtain an identity matrix:

$AB={\begin{bmatrix}1\times 5+2\times -2&1\times -2+2\times 1\\2\times 5+5\times -2&2\times -2+5\times 1\end{bmatrix}}={\begin{bmatrix}5-4&-2+2\\10-10&-4+5\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$

Similarly, on multiplying B with A, we obtain the same identity matrix:

$BA={\begin{bmatrix}5\times 1+-2\times 2&5\times 2+-2\times 5\\-2\times 1+1\times 2&2\times -2+1\times 5\end{bmatrix}}={\begin{bmatrix}5-4&10-10\\-2+2&-4+5\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$

We can that $AB=BA=I$

Hence $A^{-1}=B$ and $B$ is known as the inverse of $A$

$B^{-1}=A$ and $A$ can also be called an inverse of $B$

Invertible Matrices Theorem

Theorem 1

If there exists an inverse of a square matrix, it is always unique.

Proof:

Let $A$ is a square matrix of order $n\times n$. Let matrices $B$ and $C$ to be inverses of matrix $A$.

Now $AB=BA=I$ since $B$ is the inverse of matrix $A$.

Similarly, $AC=CA=I$.

But $B=BI=B(AC)=(BA)C=IC=C$

This proves $B=C$ or $B$ and $C$ are the same matrices.

Theorem 2

If $A$ and $B$ are matrices of the same order and are invertible, then $(AB)^{-1}=B^{-1}A^{-1}$

Proof:

As per the definition of inverse of a matrix

$(AB)(AB)^{-1}=I$

$A^{-1}(AB)(AB)^{-1}=A^{-1}I$ --------- Multiply by $A^{-1}$ on both sides

$(A^{-1}A)B(AB)^{-1}=A^{-1}$ --------- We know that $A^{-1}I=A^{-1}$    

$IB(AB)^{-1}=A^{-1}$ ---------We know that $A^{-1}A=I$  

$B(AB)^{-1}=A^{-1}$---------We know that $IB=B$  

$B^{-1}B(AB)^{-1}=B^{-1}A^{-1}$ --------- Multiply by $B^{-1}$ on both sides

$I(AB)^{-1}=B^{-1}A^{-1}$---------We know that $B^{-1}B=I$  

$(AB)^{-1}=B^{-1}A^{-1}$---------We know that $I(AB)^{-1}=(AB)^{-1}$  

Applications of Invertible Matrix

• Invertible matrices can be used to encrypt a message. There are many ways to encrypt a message and the use of coding has become particularly significant in recent years.
• Invertible matrices are employed by cryptographers to decode a message as well, especially those programming the specific encryption algorithm.
• Computer graphics in the 3D space use invertible matrices to render what you see on the screen.