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JEE Main 202628 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
28 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Matrices

JEE Main 2026 · 28 January · Evening Shift · Mathematics · Matrices

Let A= [ array ll 3 & -4 1 & -1 array ] and B be two matrices such...

Let A=A=\left[3−41−1\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]andandBbetwomatricessuchthatbe two matrices such thatA^{100}=100 B+I.Thenthesumofalltheelementsof. Then the sum of all the elements of\mathrm{B}^{100} is ____\_\_\_\_

Correct answer

0

Step-by-step solution

A=I+[2−41−2]A = I + \begin{bmatrix} 2 & -4 \\ 1 & -2 \end{bmatrix}, let M=[2−41−2]M = \begin{bmatrix} 2 & -4 \\ 1 & -2 \end{bmatrix}
M2=[0000]=M3=M4=…=M100M^2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = M^3 = M^4 = \ldots = M^{100}
A100=(I+M)100=∑r=0100(100r)Mr⋅IA^{100} = (I + M)^{100} = \sum_{r=0}^{100} \binom{100}{r} M^r \cdot I
A100=I+100M=I+100BA^{100} = I + 100M = I + 100B
∴M=B⇒M100=B100=[0000]\therefore M = B \Rightarrow M^{100} = B^{100} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
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