If A = [ * 20 c & - & ]and B = [ * 20 c & - & ], then the correct… - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{ - \sin \alpha }\\{\sin \alpha }&{\cos \alpha }\end{array}} \right]$and $B = \left[ {\begin{array}{*{20}{c}}{\cos \beta }&{ - \sin \beta }\\{\sin \beta }&{\cos \beta }\end{array}} \right]$, then the correct relation is

A
${A^2} = {B^2}$
B
$A + B = B - A$
✓
$AB = BA$
D
None of these

✓

Answer

Correct option: C.

$AB = BA$

c
(c) Clearly, $AB = \left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{ - \sin \alpha }\\{\sin \alpha }&{\cos \alpha }\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\cos \beta }&{ - \sin \beta }\\{\sin \beta }&{\cos \beta }\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}{\cos (\alpha + \beta )}&{ - \sin (\alpha + \beta )}\\{\sin (\alpha + \beta )}&{\cos (\alpha + \beta )}\end{array}} \right] = BA$ (verify).

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