5.4 Inverse of a Matrix
5.4.1 Motivation
Solving the linear two–by–two
system Eq. (
5.1
) for the components
,
of the vector
, is equivalent to the matrix equation
We recognise that in order to explicitely
solving this for
, we have to
invert the operation
.
5.4.2 Definition and Theorem
The
inverse
of a two–by–two matrix
is defined as the matrix fulfilling
The
determinant
of a two–by–two matrix
is defined as
Theorem Consider the
two–by–two matrix
If the determinant of
is non–zero, i.e.
, the inverse of
exists and is given by
For the proof of this, we just multiply
with
and
with
:
Exercise: Check the same for
.
Examples
Solving the Linear Equations
(5.1Two–by–Two Matrices)
We are now in a position to solve Eq. (
5.1
) by the inverse of a matrix: