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^-1A = I. For example, the inverse of

is

because

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:

1. 
   

2. 
   

3. 
   

4. 
   

5. 
   

6. 
   

Thus, the multiplicative inverse of

is