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 rowechelon
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