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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-1A = 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:

.    


  1.    

  2.    

  3.    

  4.    

  5.    

  6.    

Thus, the multiplicative inverse of

   

is

.    

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