Some Online Resources for This Topic:
To understand this tutorial you need to first understand matrix multiplication.
Q OK. Now we can add, subtract and multiply matrices. That leaves one more operation: divsion. Specifically, if A and B are, say, nn matrices, what is A/B?
A First ask yourself what division in the realm of real numbers really means: it is really a form of multiplication: dividing 3 by 7 is the same as multiplying 3 by 1/7, the inverse of 7. In other words,
7 
=  3  ^{.}  7 
=  3  ^{.}  7^{1} 
Thus, in the realm of real numbers, we could actually forget all about division and just multiply by the inverse whenever we wanted to divide.
Q Why complicate things like this? just tell us how to divide matrices and be done with it!
A Patience! You advocate writing down, say, 3/7 instead of 3^{.}7^{1}, which is fine (and also customary) for real numbers. But should 3/7 really mean 3^{.}7^{1} or should it mean 7^{1}^{.}3? Of course, it doesn't really matter, since multiplcation of real numbers is commutative. But multiplication of matrices is not commutative: by "B/A" should we mean A^{1}B or BA^{1}?
It is because the notion of "B divided by A" is inherently ambiguous that we never ever talk about division of matrices. Instead, we content ourselves with with using multiplicative inverses of matrices intead.
Q Fine. So how do we calculate the inverse, A^{1}, of a given matrix A?
A Not so fast! Before we try to find the inverse of a matrix A, we must first know exactly what we mean by the (multiplicative) inverse. The inverse of a number a is the number, often written a^{1}, with the property that
For example, the inverse of 76 is the number 76^{1} = 1/76, since (1/76) ^{.}76 = 76^{.} (1/76) = 1. (By the way, not every number has an inverse: the number 0 does not  it is the only number with no multiplicative inverse, and we say that zero is "not invertible.")
Now we can tell you what we mean by the inverse of a matrix:
Q Which of the following is the inverse of

Q How on earth do we find the inverses of matrices?
A If you used unknowns for all the entries of the inverse of a matrix, the condition that AA^{1} = I would give you a system of linear equations. Solving this system of equations corresponds to the following method for finding the inverse of any matrix: (See Section 3.3 in Finite Mathematics and Finite Mathematics and Applied Calculus for a detailed explanation).
Finding the Inverse of a Matrix
To determine whether the inverse A^{1} of an nn matrix A exists, and to calculate it if it does exist, follow this procedure:
Example Let
Step 1: Write down the n2n matrix [A  I]
Step 2: Row Reduce (To learn how to rowreduce a matrix, go here).
Thus,

Q What about applications? We know how to apply matrix multiplication to interesting situations. Wht about matrix inversion?
A The list of possible applications of matrix inversion is too large to mention. We mention just a few:
To learn more about these applications, consult Sections 3.3 and 3.4 of Finite Mathematics or Finite Mathematics and Applied Calculus
Comments?
Last Updated: June, 2006
Copyright © 2000, 2006 Stefan Waner