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Linear Algebra: Matrix Inverse, Left Inverse and Right Inverse
This article covers everything one need’s to know about matrix inverses. We will start by discussing the geometric interpretation of matrix inverses. Then, we’ll explain the necessary conditions for the existence of a matrix inverse, touching on concepts such as injectivity, surjectivity, and bijectivity in detail. We’ll explore why a rectangular matrix does not have an inverse but may have a left or right inverse, depending on its rank. Finally, we’ll look at how to compute the matrix inverse using elementary row operations.
Geometrical View of Inverse of Matrix
Consider a rotation matrix A that rotates a vector v by 90⁰.
A matrix B is called the inverse of matrix A if applying the linear transformation represented by B on vector w brings us back to vector v. In other words, matrix B “undoes” the transformation done by A, returning w to its original form as v.
Thus, the inverse of a matrix is the transformation that reverses the effect of the matrix.
