Download The Inverse of a matrix

The Inverse of a matrix
Let’s calculate B-1, the inverse of matrix B, just to give you an idea of what’s involved and
point out a few things. For a general 22 matrix, calculation of the inverse of a matrix is
straightforward and proceeds thusly: if
a b 
B

c d 
then its inverse is given by
B 1 
1  d  b
1  d  b



det B  c a  ad  bc  c a 
where “det B” represents the determinant of B, something you probably remember, perhaps
vaguely, from high-school algebra. Recall also that the determinant of a matrix is a scalar.
Let’s work a specific example:
1 2
B
  det B  1  4  3  2  2
3 4
1 
1  4  2   2
 B 1 



 2  3 1   3 2  1 2 
Of course, it’s always a good idea to check one’s calculations, in this case by verifying that
BB 1  I :
1    2  3 1  1  1 0
1 2  2
BB 1  
 3


I
1 
3 4  2  2   6  6 3  2 0 1
QED
But, there’s a catch, which you can appreciate if you look closely at the formula for the inverse,
1
specifically at the expression for the value of det B . Because of that
, term, not all
ad  bc
matrices can be inverted: if the determinant of a matrix is equal to zero, the matrix is termed
1
1
singular, and can’t be inverted (because
   ). For example, if we want to calculate
det( B) 0
the inverse of
3
1
Z
.
1.8 5.4
we first calculate the determinant and find that
det Z  1 5.4  1.8  3  5.4  5.4  0 ,
meaning Z is singular and we can’t invert it. We therefore can never carry out matrix division with
Z as a divisor.
Something else: multiplying a matrix by its inverse, which is the equivalent of dividing a
matrix by itself, yields the identity matrix. Try this with the matrix
1 4
A
.
3 2
First, try calculating the inverse for yourself. You should come up with
 0.2 0.4 
A 1  

 0.3  0.1
Then, if you carry out the multiplication of A by A-1, you should obtain the corresponding identity
matrix (try it).
Finally, although calculating the inverse of a 22 matrix is easy, for anything larger, it’s not.
Here’s a link to a site that will walk you through the process for a 33:
http://www.easycalculation.com/matrix/inverse-matrix-tutorial.php
That should convince you to leave matrix inverse calculations to MATLAB!