Invertible matrices are very important in many areas of science. For example, decrypting a coded message uses invertible matrices (see the coding page). The problem of finding the inverse of a matrix will be discussed in a different page (click here).
matrix A is called nonsingular or invertible iff there exists an
matrix B such that
Notation. A common notation for the inverse of a matrix A is A-1. So
Example. Find the inverse of
The inverse matrix is unique when it exists. So if A is invertible, then A-1 is also invertible and
The following basic property is very important:
Remark. In the definition of an invertible matrix A, we used both and to be equal to the identity matrix. In fact, we need only one of the two. In other words, for a matrix A, if there exists a matrix B such that , then A is invertible and B = A-1.
More on invertible matrices and how to find the inverse matrices will be discussed in the Determinant and Inverse of Matrices page.
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Author: M.A. Khamsi