Understanding Matrices | Part 4: Matrix Inverse

A matrix inverse is defined as a reverse transformation that maps an output vector back to its original input vector. An inverse exists only if the original transformation is one-to-one, meaning no two different inputs produce the same output. The inverse of a product of matrices, such as (AB)⁻¹, is the product of their inverses in reverse order, B⁻¹A⁻¹, because the transformations must be undone in the reverse sequence they were applied. The article also explains that special matrices, like cyclic-shift or exchange matrices, have simple inverses that are often their own transpose or the matrix itself.