If ω is a complex cube root of unity and A=[ beginmatrix ω 0 0 ω endmatrix ], then A50 is

Question

If ω is a complex cube root of unity and A=[ beginmatrix ω 0 0 ω endmatrix ], then A50 is (A)  ω 2A  (B)  ω A  (C)  A  (D)  0

Tardigrade Admin · Accepted Answer

Given, A=[ beginmatrix ω 0 0 ω endmatrix ] A2=[ beginmatrix ω 0 0 ω endmatrix ][ beginmatrix ω 0 0 ω endmatrix ]=[ beginmatrix ω 2 0 0 ω 2 endmatrix ] A3=[ beginmatrix ω 2 0 0 ω 2 endmatrix ][ beginmatrix ω 0 0 ω endmatrix ]=[ beginmatrix ω 3 0 0 ω 3 endmatrix ] Similarly, A50=[ beginmatrix ω 50 0 0 ω 50 endmatrix ] =[ beginmatrix (ω 3)16ω 2 0 0 (ω 3)16ω 2 endmatrix ] =[ beginmatrix ω 2 0 0 ω 2 endmatrix ]=ω A

Q. If $ω$ is a complex cube root of unity and $A = [ω 0 0 ω],$ then $A^{50}$ is

$ω^{2} A$

$ω A$

$A$

$0$

Given, $A = [ω 0 0 ω]$
$A^{2} = [ω 0 0 ω] [ω 0 0 ω] = [ω^{2} 0 0 ω^{2}]$
$A^{3} = [ω^{2} 0 0 ω^{2}] [ω 0 0 ω] = [ω^{3} 0 0 ω^{3}]$
Similarly, $A^{50} = [ω^{50} 0 0 ω^{50}]$
$= [(ω^{3})^{16} ω^{2} 0 0 (ω^{3})^{16} ω^{2}]$
$= [ω^{2} 0 0 ω^{2}] = ω A$

Q. If ω is a complex cube root of unity and A=[ω0​0ω​], then A50 is

ω2A

ωA

A

0

Given, A=[ω0​0ω​] A2=[ω0​0ω​][ω0​0ω​]=[ω20​0ω2​] A3=[ω20​0ω2​][ω0​0ω​]=[ω30​0ω3​] Similarly, A50=[ω500​0ω50​] =[(ω3)16ω20​0(ω3)16ω2​] =[ω20​0ω2​]=ωA

Q. If $ω$ is a complex cube root of unity and $A = [ω 0 0 ω],$ then $A^{50}$ is

$ω^{2} A$

$ω A$

$A$

$0$

Given, $A = [ω 0 0 ω]$
$A^{2} = [ω 0 0 ω] [ω 0 0 ω] = [ω^{2} 0 0 ω^{2}]$
$A^{3} = [ω^{2} 0 0 ω^{2}] [ω 0 0 ω] = [ω^{3} 0 0 ω^{3}]$
Similarly, $A^{50} = [ω^{50} 0 0 ω^{50}]$
$= [(ω^{3})^{16} ω^{2} 0 0 (ω^{3})^{16} ω^{2}]$
$= [ω^{2} 0 0 ω^{2}] = ω A$