Understanding the Inverse of a Matrix: Properties and Non-Invertibility

Introduction

The inverse of a matrix is a crucial concept in linear algebra, playing a vital role in solving systems of linear equations, transforming coordinate systems, and other areas of mathematics. This article provides a comprehensive overview of the inverse of a square matrix, explaining when a matrix has an inverse and when it does not, along with the properties of matrices that cannot have inverses.

What is the Inverse of a Matrix?

A square matrix $A$ is said to have an inverse, denoted as $A^{-1}$, if there exists another square matrix of the same size such that $A A^{-1} A^{-1} A I_n$, where $I_n$ is the identity matrix of order $n$. The existence of an inverse is contingent on the matrix being nonsingular (i.e., its determinant is non-zero).

Definition of the Inverse

The unique inverse of a matrix $A$ can be defined as follows:

Definition. Let $A$ be a square matrix of order $n$ over a field $F$. Then another matrix $B$ of the same size is the inverse of $A$ if $A B B A I_n$: the identity matrix of order $n$: $I_n delta_{ij}$, where $delta_{ij}$ is the Kronecker delta.

The uniqueness of the inverse can be proven as follows: Assume there exist two different matrices $C$ and $C'$ such that $A C C' A I_n$. From the definition of the identity matrix, we get:

$C C I_n C (A C') (C A) C' I_n C' C'$.

This demonstrates that any matrix satisfying the condition for being an inverse must be equal to the unique inverse of $A$.

When a Matrix Does Not Have an Inverse

A square matrix $A$ does not have an inverse if it is singular, which means its determinant is zero, or in other words, $A 0$. In such cases, the matrix is termed singular or non-invertible.

Examples of Non-Invertible Matrices

In some non-square matrices, both $A B$ and $B A$ are not defined or equal to different sizes. Only in the case of square matrices can both $A B$ and $B A$ be the identity matrix if they exist.

Matrix $A$ has a zero row or column. This can be represented as $A_i 0 in F^n$.

Matrix $A$ has at least two proportional rows or columns. This can be represented as $A_i lambda A_k$ or $A^j mu A^m$ with $lambda eq 0$ and $mu eq 0$.

Matrix $A$ has a subset of linearly dependent rows or columns. This encompasses the cases mentioned above.

In these cases, the matrix $A$ is non-invertible, and there can be infinitely many such matrices depending on the specific properties and conditions that make them singular.

Conclusion

The concept of the inverse matrix is fundamental in linear algebra, and understanding when a matrix has an inverse is essential. By examining the determinant and the row and column properties, we can determine the invertibility of a matrix. Non-invertible matrices, while not having an inverse, can still provide valuable insights into the structure and properties of the matrix within the context of various mathematical and computational tasks.

References

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985. A. Carausu, Linear Algebra - Theory and Applications, Matrix Rom Publishers, Bucharest, 1999.