4Matrices and linear equations
IA Vectors and Matrices
4.1 Simple example, 2 × 2
Consider the system of equations
A
11
x
1
+ A
12
x
2
= d
1
(a)
A
21
x
1
+ A
22
x
2
= d
2
. (b)
We can write this as
Ax = d.
If we do (a)×A
22
−(b)×A
12
and similarly the other way round, we obtain
(A
11
A
22
− A
12
A
21
)x
1
= A
22
d
1
− A
12
d
2
(A
11
A
22
− A
12
A
21
)
| {z }
det A
x
2
= A
11
d
2
− A
21
d
1
Dividing by det A and writing in matrix form, we have
x
1
x
2
=
1
det A
A
22
−A
12
−A
21
A
11
d
1
d
2
On the other hand, given the equation
Ax
=
d
, if
A
−1
exists, then by multiplying
both sides on the left by A
−1
, we obtain x = A
−1
d.
Hence, we have constructed
A
−1
in the 2
×
2 case, and shown that the
condition for its existence is det A 6= 0, with
A
−1
=
1
det A
A
22
−A
12
−A
21
A
11