Matrices
The Inverse of a Matrix
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
A = 
|
is
A
-1 = 
|
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:
 .
|
|
|
is
 .
|
