Let A = [ 3 - 4 1 - 1 ] and B be two matrices such that A 100 = 100 B + I Then the sum of all the elements of B 100 is ________. [2026]

Question

Let $A = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ and B be two matrices such that $A^{100} = 100 B + I$ . Then the sum of all the elements of $B^{100}$ is ________. [2026]

Smart Achievers · Accepted Answer

(0)

$A = I + [\begin{matrix} 2 & - 4 \\ 1 & - 2 \end{matrix}], let M = [\begin{matrix} 2 & - 4 \\ 1 & - 2 \end{matrix}]$

$M^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \Rightarrow M^{3} = M^{4} = \dots = M^{100} = 0$

$A^{100} = {(I + M)}^{100} = \sum_{r = 0}^{100} (\binom{100}{r}) M^{r} . I$

$A^{100} = I + 100 M = I + 100 B$

$∴$ $M = B \Rightarrow M^{100} = B^{100} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

Solution

Q.
Let $A = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ and B be two matrices such that $A^{100} = 100 B + I$ . Then the sum of all the elements of $B^{100}$ is ________. [2026]

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Solution

Q. Let A=[3-41-1] and B be two matrices such that A100=100B+I. Then the sum of all the elements of B100 is ________. [2026]

Ans. (0) A=I+[2-41-2], let M=[2-41-2] M2=[0000]⇒M3=M4=⋯=M100=0 A100=(I+M)100=∑r=0100(100r)Mr.I A100=I+100M=I+100B ∴ M=B⇒M100=B100=[0000]

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Q.
Let $A = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ and B be two matrices such that $A^{100} = 100 B + I$ . Then the sum of all the elements of $B^{100}$ is ________. [2026]