Download I n

The Inverse of a Matrix
(10/14/05)
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If A is a square (say n by n) matrix and
if there is an n by n matrix C such that
C A = A C = In , then C is called the
inverse of A, is denoted A -1, and A is
said to be an invertible matrix.
Viewed as functions from Rn to Rn , this
says that A and A -1 are inverse
functions of each other (since In is the
identity function).
The 2 by 2 Case
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If A = a b  then the determinant of A
c d 
is the number a d – b c . This is denoted
det(A).
It is easy to check that:

If det(A)  0 then the inverse of A is
d  b


c
a


1/det(A) 

If det(A) = 0, then A is not invertible.
Using Inverses
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If A is an invertible (hence square) matrix
and if we wish to solve the matrix
equation A x = b (x and b vectors in Rn),
then x = A -1 b .
That is, you can either solve the equation
by our standard method, or you can
compute A -1 and multiply b by it.
In general, the former way is less work
(but not always).
Computing Inverses

An algorithm for computing A


-1
is:
Write down the n by 2n augmented matrix
[A In ]
Do row reduction until the A half is in
reduced echelon form. If the left half is
now In , then the matrix on the right is A -1.
If the left half is not In (i.e., not a pivot in
every row/column), then A is not invertible.
The Invertible Matrix Theorem
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See Page 129 of our text.
The theorem is a list of equivalent ways
of saying that a given matrix A is
invertible. It basically boils down to the
fact that A is invertible if and only if as
a linear transformation from Rn to Rn
A is one-to-one (and hence also onto).
Assignment for Monday
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Catch up on any problems you have left
behind in Chapter 2 so far.
Read Section 2.3.
Do the Practice and Exercises 1–7 odd,
11, 13 and 19.