# Invertible Matrices

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).

Definition. An matrix A is called nonsingular or invertible iff there exists an matrix B such that

where In is the identity matrix. The matrix B is called the inverse matrix of A.

Example. Let

One may easily check that

Hence A is invertible and B is its inverse.

Notation. A common notation for the inverse of a matrix A is A-1. So

Example. Find the inverse of

Write

Since

we get

Easy algebraic manipulations give

or

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:

If A and B are invertible matrices, then is also invertible and

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