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

**Example.** Let

One may easily check that

Hence

**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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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

**Author**: M.A. Khamsi

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