The matrices R = [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]] and D =…
1996
The matrices R = [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]] and D = [[a, 0], [0, b]] commute under multiplication
- A.
if a = b or theta = n*pi, where n is an integer
- B.
always
- C.
never
- D.
if a cos(theta) = b sin(theta)
Attempted by 9 students.
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Correct answer: A
Let R = [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]] and D = [[a, 0], [0, b]].
Multiplying, R D = [[a cos(theta), -b sin(theta)], [a sin(theta), b cos(theta)]], while D R = [[a cos(theta), -a sin(theta)], [b sin(theta), b cos(theta)]].
For R D = D R, the off-diagonal entries must match, so -b sin(theta) = -a sin(theta) and a sin(theta) = b sin(theta). Hence (a - b) sin(theta) = 0. Therefore either a = b or sin(theta) = 0, i.e. theta = n*pi for some integer n. Thus option 1 is correct.