## 3.3 Matrix Inversion

(This topic is also in Section 3.3 in Finite Mathematics and Finite Mathematics & Applied Calculus) Some On-line 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, n n 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,

 3 7 = 3 . 1 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-1B 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

a-1.a = a.a-1 = 1.

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:

Inverse of a Matrix

The inverse of an n n matrix A is that n n matrix A-1 which, when multiplied by A on either side, yields the n n identity matrix I. Thus,

AA-1 =A-1A = I.

If A has an inverse, it is called invertible. Otherwise, it is called singular.

Examples

1. The inverse of the 1 1 matrix  is [1/3], since

[1/3] =  = [1/3]

2. The inverse of the n n identity matrix I is I itself, since

I I = I.
Thus, I-1 = I.

3. Here is an inverse of a 2 2 matrix: 1 -1 -1 -1
-1 = 1 2 - 1 2 - 1 2 - 1 2

Q Why?
A Because 1 -1 -1 -1 1 2 - 1 2 - 1 2 - 1 2
= 1 0 0 1
A A-1 = I
Also, 1 2 - 1 2 - 1 2 - 1 2 1 -1 -1 -1
=   Q Which of the following is the inverse of 2 3 1 2
?

Q Let

A= 1 0 -2 4 1 0 1 1 7
Then
A-1= -2 2 -28 9 -8 3 -1 1 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 A A-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 n n matrix A exists, and to calculate it if it does exist, follow this procedure:

1. Write down the n 2n matrix [A | I] (this is A with the n�n identity matrix set next to it).
2. Row-reduce [A | I].
3. If the reduced form is [I | B] (i.e., has the identity matrix in the left part), then A is invertible and B = A-1. If you cannot obtain I in the left part, then A is singular.

Example

Let

A= 1 2 -3 -4

Step 1: Write down the n 2n matrix [A | I]
This gives us

[A | I]= 1 2 1 0 -3 -4 0 1 A I

Step 2: Row Reduce (To learn how to row-reduce a matrix, go here). 1 2 1 0  Reduce 1 0 -2 -1 -3 -4 0 1 0 1 3 2 1 2 I A-1

Thus,

A-1= -2 -1 3 2 1 2 Let   A= 1 0 0 2 -1 0 1 1 1

Q To compute the inverse, the matrix you must reduce is (use the "tab" key to move from cell to cell):

[A | I]=  Q What matrix do you obtain if you completely row-reduce the above matrix? (It is good practice to reduce the matrix by hand, as in the Section 2.2 tutorial. If you are feeling lazy and want to use technology, go here.)  Q It follows that

A-1=  Now let A= 3 0 1 2 -1 0 1 1 1

Q Which of the following is true? 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:

1. Solving systems of n linear equations in n unknowns (a mathematical application)
2. Brand switching by customers (a business application)
3. Input-Output Models (Next tutorial)

To learn more about these applications, consult Sections 3.3 and 3.4 of Finite Mathematics or Finite Mathematics and Applied Calculus 