Let A=0α00 , I=1001 and (A + I)50 -50A=abcd, the value of a + b + c + d, is

we are given that A=0α00and (A+I)50-50A=abcd→(1)Hence A+I=0α00+1001=1α01 (A+I)2=(A+I) (A+I)=1α011α01=12α01 (A+I)3=(A+I)2+(A+I)=12α011α01=13α01 Similarly,(A+I)50=150α01Putting above values in (1), we get⇒150α01-050α00=abcd ⇒1001=abcd ⇒a=1,b=0,c=0,d=1 ⇒a+b+c+d=1+0+0+1=2Hence option (b) is correct

Let A=0α00 , I=1001 and (A + I)50 -50A=abcd, the value of a + b + c + d, is

we are given that A=0α00and (A+I)50-50A=abcd→(1)Hence A+I=0α00+1001=1α01 (A+I)2=(A+I) (A+I)=1α011α01=12α01 (A+I)3=(A+I)2+(A+I)=12α011α01=13α01 Similarly,(A+I)50=150α01Putting above values in (1), we get⇒150α01-050α00=abcd ⇒1001=abcd ⇒a=1,b=0,c=0,d=1 ⇒a+b+c+d=1+0+0+1=2Hence option (b) is correct

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Let $A = [\begin{matrix} 0 & α \\ 0 & 0 \end{matrix}]$ , $I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and ${(A + I)}^{50} - 50 A = [\begin{matrix} a & b \\ c & d \end{matrix}],$ the value of $a + b + c + d,$ is

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None of the above

answer is B.

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Detailed Solution

we are given that $A = [\begin{matrix} 0 & α \\ 0 & 0 \end{matrix}]$

and ${(A + I)}^{50} - 50 A = [\begin{matrix} a & b \\ c & d \end{matrix}] \to (1)$

Hence $A + I = [\begin{matrix} 0 & α \\ 0 & 0 \end{matrix}] + [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & α \\ 0 & 1 \end{matrix}] {(A + I)}^{2} = (A + I) (A + I) = [\begin{matrix} 1 & α \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & α \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 2 α \\ 0 & 1 \end{matrix}] {(A + I)}^{3} = {(A + I)}^{2} + (A + I) = [\begin{matrix} 1 & 2 α \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & α \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 3 α \\ 0 & 1 \end{matrix}]$

Similarly, ${(A + I)}^{50} = [\begin{matrix} 1 & 50 α \\ 0 & 1 \end{matrix}]$

Putting above values in (1), we get

$\Rightarrow [\begin{matrix} 1 & 50 α \\ 0 & 1 \end{matrix}] - [\begin{matrix} 0 & 50 α \\ 0 & 0 \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] \Rightarrow [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] \Rightarrow a = 1, b = 0, c = 0, d = 1 \Rightarrow a + b + c + d = 1 + 0 + 0 + 1 = 2$