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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.

Layla Smith
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    The most common application is solving systems of linear equations. See: https://en.wikipedia.org/wiki/System_of_linear_equations#Matrix_equation – Augustin Aug 16 '15 at 16:12
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    In the first place, solving matrix equations. In the second place it is very important for solving linear systems of differential equations that the exponential of a matrix be invertible. – Bernard Aug 16 '15 at 16:12
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    Compare the scalar equation, $ax = b$, to the matrix equation, $Ax=b$. In both cases, you can solve for the unknown, $x$, by multiplying by the inverse of $a$ or $A$ on the left (assuming $A^{-1}$ exists). – Chester Aug 16 '15 at 16:15
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    In linear regression modelling, least squares estimates rely upon invertible matrices to acquire precision for parameter estimates. – miradulo Aug 16 '15 at 16:15

1 Answers1

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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.

Hirshy
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