Math 2331
Linear Algebra
Sections 2.5
Inverses of Matrices
Recall the following basic facts of real number multiplication:
The number 1 is called the multiplicative identity since
1*
*1
x
x
x
=
=
for all real
numbers x.
The numbers a and b are called multiplicative inverses if
*
1
a
b
=
In matrix multiplication the identity matrix is called I. Note that we use the generic term I
to indicate an identity, you need the context to determine the size of the square matrix I.
For all matrices
A
*
I
=
A
m
n
×
m
n
n
n
×
×
I
*
A
=
A
m
m
×
m
n
m
n
×
×
Two square matrices A and B are called inverses of each other if
AB = BA = I
Given the matrix A, its inverse is denoted
1
A
−
Properties of Inverses
1. A square n x n matrix has an inverse EXACTLY when it has a full set of n pivots.
Notice that when solving Ax = b, it had a unique solution when A had a full set of pivots.
If A has an inverse,
1
1
A
Ax
A b
−
−
=
If A doesn't have a full set of pivots, we can find a nonzero solution to Ax = 0
Suppose A had an inverse but not a full set of pivots
Ax = 0

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