Let M =[0 -α α 0], where α is a non-zero real number an N = displaystyle∑ k =149 M 2 k . If ( I - M 2) N =-2 I, then the positive integral value of α is

Question

Let M =[0 -α α 0], where α is a non-zero real number an N = displaystyle∑ k =149 M 2 k . If ( I - M 2) N =-2 I, then the positive integral value of α is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

M =[0 -α α 0] ; M 2=[-α2 0 0 -α2]=-α2 I N = M 2+ M 4+ ldots ldots+ M 98=[-α2+α4-α6+ ldots] I =-α2 ((1-(-α2)49)/1+α2) . I I - M 2=(1+α2) I ( I - M 2) N =-α2(α98+1)=-2 α=1

Q. Let $M = [0 α - α 0]$ , where $α$ is a non-zero real number an $N = k = 1 \sum 49 M^{2 k}$ . If $(I - M^{2}) N = - 2 I$ , then the positive integral value of $α$ is______

$M = [0 α - α 0]; M^{2} = [- α^{2} 0 0 - α^{2}] = - α^{2} I$
$N = M^{2} + M^{4} + \dots\dots + M^{98} = [- α^{2} + α^{4} - α^{6} + \dots] I$
$= - α^{2} \frac{( 1 - ( - α ^{2} ) ^{49} )}{1 + α ^{2}} . I$
$I - M^{2} = (1 + α^{2}) I$
$(I - M^{2}) N = - α^{2} (α^{98} + 1) = - 2$
$α = 1$

Q. Let M=[0α​−α0​], where α is a non-zero real number an N=k=1∑49​M2k. If (I−M2)N=−2I, then the positive integral value of α is______

M=[0α​−α0​];M2=[−α20​0−α2​]=−α2I N=M2+M4+……+M98=[−α2+α4−α6+…]I =−α21+α2(1−(−α2)49)​.I I−M2=(1+α2)I (I−M2)N=−α2(α98+1)=−2 α=1

Q. Let $M = [0 α - α 0]$ , where $α$ is a non-zero real number an $N = k = 1 \sum 49 M^{2 k}$ . If $(I - M^{2}) N = - 2 I$ , then the positive integral value of $α$ is______

$M = [0 α - α 0]; M^{2} = [- α^{2} 0 0 - α^{2}] = - α^{2} I$
$N = M^{2} + M^{4} + \dots\dots + M^{98} = [- α^{2} + α^{4} - α^{6} + \dots] I$
$= - α^{2} \frac{( 1 - ( - α ^{2} ) ^{49} )}{1 + α ^{2}} . I$
$I - M^{2} = (1 + α^{2}) I$
$(I - M^{2}) N = - α^{2} (α^{98} + 1) = - 2$
$α = 1$