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.
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.
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