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The Inverse of a Matrix

The multiplicative inverse of a
real number is the number that yields 1 (the identity) when multiplied
by the original number. is the multiplicative inverse of
*a*, because *a*× = 1.

Most matrices also have a multiplicative inverse. In other words, for
the majority of matrices *A*, there exists a matrix *A*^{-1} such that
*AA*^{-1} = *I* and *A*^{-1}*A* = *I*. For example, the
inverse of

is

because

and

To find the inverse of an *m*×*m* matrix, write the *m*×*m*
matrix on the left,
and the *m*×*m* identity matrix to the right:

Then, row reduce to convert the matrix to reduced row-echelon
form—that is, to get an

*m*×*m* identity matrix on the left.
The new

*m*×*m* matrix on the right is the
multiplicative inverse of the original matrix. In other words, the new

*m*×*m* matrix times the original

*m*×*m* matrix yields the
identity matrix.

*Example*: Find the multiplicative inverse of:

Thus, the multiplicative inverse of

is