# I dont get how these matrices are mulitiplied together

1. Feb 5, 2012

### cs23

1. The problem statement, all variables and given/known data
H10 =
c1 | -s1 | 0
s1 | c1 | 0
0 | 0 | 1

H21 =
c2 | -s2 | 0
s2 | c2 | 0
0 | 0 | 1

2. Relevant equations
this is what i did

H10H21 =
c1c2 -s1s2 | -c1s2 - s1c2 | 0
s1c2 + c1s2 | -s1s2 + c1c2 | 0
0 | 0 | 1

3. The attempt at a solution

H10H21 =

c1c2 | -s1s2 | 0
s1s2 | c1c2 | 0
0 | 0 | 1

Last edited: Feb 5, 2012
2. Feb 5, 2012

### HallsofIvy

Unless that is some other information that you have not given, your answer is correct.
Note, by the way, that if $c_1^2+ s_1^2= 1$ and $c_2^2+ s_2^2= 1$, or if you divide each term of the matrices by that, we can interpret $c_1$ and $s_1$ as the cosine and sine of some angle, $\theta$ and can interpret $c_2$ and $s_2$ as the cosine and sine of some other angle, $\phi$ and so these two matrices as rotation, about the z-axis, through those angles. The product would be the combination of those two rotations, a rotation through angle $\theta+ \phi$ and then we have $c_1c_2- s_1s_2= cos(\theta)cos(\phi)- sin(\theta)sin(\phi)= cos(\theta+ \phi)$ and $s_1c_1+ c_1s_2= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)= sin(\theta+ \phi)$ as we should have.

(What you give as the "answer" is sometimes called the "component by component" product but it does NOT have any good algebraic properties and is very seldom used.)