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