Question

$IfA=\left[\begin{array}{cc}co{s}^2\alpha & cos\alpha sin\alpha \\ cos\alpha sin\alpha & si{n}^2\alpha \end{array}\right]andB=\left[\begin{array}{cc}co{s}^2\beta & cos\beta sin\beta \\ cos\beta sin\beta & si{n}^2\beta \end{array}\right]$ are two matrices such that AB is the null matrix, then

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$AB=0\Rightarrow \left[\begin{array}{cc}co{s}^2\alpha & cos\alpha sin\alpha \\ cos\alpha sin\alpha & si{n}^2\alpha \end{array}\right]\left[\begin{array}{cc}co{s}^2\beta & cos\beta sin\beta \\ cos\beta sin\beta & si{n}^2\beta \end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow \left[\begin{array}{cc}cos\alpha cos\beta cos(\alpha -\beta )& cos\alpha sin\beta cos(\alpha -\beta )\\ cos\beta sin\alpha cos(\alpha -\beta )& sin\alpha sin\beta cos(\alpha -\beta )\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow cos(\alpha -\beta )=0$