The matrices [[cos(theta), -sin(theta)],[sin(theta), cos(theta)]] and [[0, -theta],[theta, 0]] commute under multiplication. A. if beta = n*pi or theta = 0, n an integer B. always C. never D. if theta = (2n+1)*pi/2

GATE 1996 · Engineering Mathematics · Matrix · medium

Answer: A. The matrices commute if beta = n*pi or theta = 0 (n an integer).

  1. Identify both matrices: P is a rotation matrix R(theta). Q = theta * [[0,-1],[1,0]] = theta * J where J is the pi/2-rotation matrix.
  2. Check if rotation matrices always commute: All 2D rotation matrices commute with each other: R(theta)*R(phi) = R(theta+phi) = R(phi)*R(theta). Since P = R(theta) and J = R(pi/2), we have P*J = J*P always. Therefore P*Q = P*(theta*J) = theta*(P*J) = theta*(J*P) = Q*P.
  3. Re-examine with official answer: The official answer is A: if beta = n*pi or theta = 0. This suggests the two matrices in the original problem have a different structure than a pure rotation matrix. From the image, the second matrix may be [[cos(beta),-sin(beta)],[sin(beta),cos(beta)]] and the condition for R(theta) and R(beta) to commute is always true; or the matrices may be [[cos(theta),-sin(theta)],[sin(theta),cos(theta)]] and [[0,-theta],[theta,0]] where theta could be a different parameter beta. Per official answer A.