If A=[ beginmatrix cos α sin α - sin α cos α endmatrix ], then A2 is equal to

Question

If A=[ beginmatrix cos α sin α - sin α cos α endmatrix ], then A2 is equal to (A)  [ beginmatrix cos 2α sin 2α - sin 2α cos 2α endmatrix ]  (B)  [ beginmatrix cos 2α - sin 2α - sin 2α cos 2α endmatrix ]  (C)  [ beginmatrix sin 2α cos 2α - cos 2α sin 2α endmatrix ]  (D)  [ beginmatrix 1 0 0 1 endmatrix ]

Tardigrade Admin · Accepted Answer

A=[ beginmatrix cos α sin α - sin α cos α endmatrix ] A2=A.A =[ beginmatrix cos α sin α - sin α cos α endmatrix ][ beginmatrix cos α sin α - sin α cos α endmatrix ] =[ beginmatrix cos 2α - sin 2α cos α sin α + sin α cos α - sin α cos α - sin α cos α - sin 2α + cos 2α endmatrix ] =[ beginmatrix cos 2α sin 2α - sin 2α cos 2α endmatrix ]

Q. If $A = [cos α - sin α sin α cos α],$ then $A^{2}$ is equal to

$[cos 2 α - sin 2 α sin 2 α cos 2 α]$

$[cos^{2} α - sin^{2} α - sin^{2} α cos^{2} α]$

$[sin^{2} α - cos^{2} α cos^{2} α sin^{2} α]$

$[1001]$

$A = [cos α - sin α sin α cos α]$
$A^{2} = A . A$
$= [cos α - sin α sin α cos α] [cos α - sin α sin α cos α]$
$= [cos^{2} α - sin^{2} α - sin α cos α - sin α cos α cos α sin α + sin α cos α - sin^{2} α + cos^{2} α]$ $= [cos 2 α - sin 2 α sin 2 α cos 2 α]$

Q. If A=[cosα−sinα​sinαcosα​], then A2 is equal to

[cos2α−sin2α​sin2αcos2α​]

[cos2α−sin2α​−sin2αcos2α​]

[sin2α−cos2α​cos2αsin2α​]

[10​01​]