Question

The matrices $\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]and\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$commute under mulitpication.

• A. $ifa=b$ or $\theta =n\pi$, n is an integer
• B. Always
• C. Never
• D. $ifacos\theta \ne bsin\theta$

Answer 1

The correct option is A $ifa=b$ or $\theta =n\pi$, n is an integer

Given
$A=\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]$
and $B=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
=$\left[\begin{array}{cc}acos\theta & -bsin\theta \\ asin\theta & bcos\theta \end{array}\right]$
$BA=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]$
$=\left[\begin{array}{cc}acos\theta & -asin\theta \\ bsin\theta & bcos\theta \end{array}\right]$
Now AB=BA
$\Rightarrow \left[\begin{array}{cc}acos\theta & -bsin\theta \\ asin\theta & bcos\theta \end{array}\right]=\left[\begin{array}{cc}acos\theta & -asin\theta \\ bsin\theta & bcos\theta \end{array}\right]$
On comparison
$asin\theta =bsin\theta$
(a-b)$sin\theta =0$
So either a=b or $\theta =n\pi$
(i.e. A and B commute when a=b (or) $\theta =n\pi$ is an integer .