Linear Algebra

Section IV. Matrix Operations 235
This procedure will find the inverse of a general n×n matrix. The 2×2 case
is handy.
4.12 Corollary The inverse for a 2×2 matrix exists and equals
if and only if ad − bc �= 0.
� �−1 a b
=
c d
1
ad − bc
�
d
�
−b
−c a
Proof. This computation is Exercise 22. QED
We have seen here, as in the Mechanics of Matrix Multiplication subsection,
that we can exploit the correspondence between linear maps and matrices. So
we can fruitfully study both maps and matrices, translating back and forth to
whichever helps us the most.
Over the entire four subsections of this section we have developed an algebra
system for matrices. We can compare it with the familiar algebra system for
the real numbers. Here we are working not with numbers but with matrices.
We have matrix addition and subtraction operations, and they work in much
the same way as the real number operations, except that they only combine
same-sized matrices. We also have a matrix multiplication operation and an
operation inverse to multiplication. These are somewhat like the familiar real
number operations (associativity, and distributivity over addition, for example),
but there are differences (failure of commutativity, for example). And, we have
scalar multiplication, which is in some ways another extension of real number
multiplication. This matrix system provides an example that algebra systems
other than the elementary one can be interesting and useful.
Exercises
4.13 Supply the intermediate steps in Example 4.10.
� 4.14 Use � Corollary � 4.12 to� decide �if
each matrix � has an � inverse.
2 1
0 4
2 −3
(a)
(b)
(c)
−1 1
1 −3
−4 6
� 4.15 For each invertible matrix in the prior problem, use Corollary 4.12 to find its
inverse.
� 4.16 Find the inverse, if it exists, by using the Gauss-Jordan method. Check the
answers for the 2×2 matrices with Corollary 4.12.
� � � � � �
3 1
2 1/2
2 −4
(a)
(b)
(c)
0 2
3 1
−1 2
�
0 1
�
5
�
2 2
�
3
(e) 0 −2 4 (f) 1 −2 −3
2 3 −2
4 −2 −3
� 4.17 What matrix has this one for its inverse?
� �
1 3
2 5
(d)
�
1 1
�
3
0 2 4
−1 1 0