Why we need Invertible Matrices In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or non degenerate) if there exists an n-by-n square matrix B such that
$$A B = B A=I_n$$
I know the definition. But what are the practical applications of of invertible matrix.
 A: The first practical application that comes to mind should be handling of matrix equations and solving of a system of linear equations, see here. If you have an equation of the form $$Ax=y$$ with $A\in\mathbb R^{n\times n},x,y\in\mathbb R^{n\times 1}$ and you know that $A$ is invertible, you can find the solution via multiplication: $$Ax=y \iff A^{-1}Ax=A^{-1}y \iff I_nx=A^{-1}y \iff x=A^{-1}y.$$
But this only deals with one application, there is much more theory to matrices to discover e.g. in linear algebra. Assume we have $\mathcal V$ and $\mathcal W$ finite-dimensional vector spaces over the same field $F$, then every linear map from $\mathcal V$ to $\mathcal W$ can be represented by a matrix $A\in F^{m\times n}$ with $\dim(\mathcal V)=n$ and $\dim(\mathcal W)=m$. 
Now let $\dim(\mathcal V)=\dim(\mathcal W)$, then we have a matrix $A\in F^{n\times n}$. If we know that $A$ is invertible, we immediately know that the corresponding linear map $\varphi: \mathcal V\rightarrow\mathcal W$ is bijective and we also know that the corresponding linear map to $A^{-1}$ is $\varphi^{-1}$. Using some properties of a linear map we also know that e.g. $0$ is not an eigenvalue of $\varphi$ nor $\varphi^{-1}$ etc. So just by having $A$ invertible we know lots of things about the corresponding linear map.
