I dont get how these matrices are mulitiplied together

[cs23]

Homework Statement

H₁⁰ =
c1 | -s1 | 0
s1 | c1 | 0
0 | 0 | 1

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

Homework Equations

this is what i did

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

The Attempt at a Solution

this is the answer

H₁⁰H₂¹ =

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

 

[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.)