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Powers of a matrix of the form I + N where N is nilpotent

This type of question, asked 9 times: 2021 Paper 17, 29 Jun 2022, 25 Jul 2022, 12 Apr 2023, 4 Apr 2024, 29 Jan 2025, 21 Jan 2026 S2, 28 Jan 2026 S2, 6 Apr 2026 S2. All 9 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    90
    If A=[111011001]A=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} and M=A+A2+A3+…..+A20M=A+A^{2}+A^{3}+ \ldots.. +A^{20}, then the sum of all the elements of the matrix MM is equal to

  2. Question 2

    29 Jun 2022

    153
    Let A=(2−102)A =\begin{pmatrix} 2 & - 1 \\ 0 & 2 \end{pmatrix}. If B=I− 5C1(adjA)+ 5C2B = I -\ ^{5}C_{1}(adjA) +\ ^{5}C_{2} (adjA)2−…− 5C5(adjA)5(adjA)^{2}- \ldots -\ ^{5}C_{5}(adjA)^{5}, then the sum of all elements of the matrix BB is:

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  3. Question 3

    25 Jul 2022

    173
    Let A=[1aa01b001],a,b∈RA =\begin{bmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix},a,b \in R. If for some n∈Nn \in N, An=[14821600196001]A^{n}=\begin{bmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{bmatrix} then n+a+bn + a + b is equal to

  4. Question 4

    12 Apr 2023

    74
    Let A=[115101]A =\begin{bmatrix} 1 & \frac{1}{51} \\ 0 & 1 \end{bmatrix}. If B=[12−1−1]A[−1−211]B =\begin{bmatrix} 1 & 2 \\ - 1 & - 1 \end{bmatrix}A\begin{bmatrix} - 1 & - 2 \\ 1 & 1 \end{bmatrix}, then the sum of all the elements of the matrix ∑n=150Bn\sum_{n= 1}^{50} B^{n} is equal to

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  5. Question 5

    4 Apr 2024

    156
    Let A=[1201]A =\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} and B=I+adj(A)+(adjA)2+…+B = I + adj(A) + (adjA)^{2}+ \ldots + (adjA)10(adjA)^{10}. Then, the sum of all the elements of the matrix B is:

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  6. Question 6

    29 Jan 2025

    72
    Let S={m∈Z:Am2+Am=3I−A−6}S =\left\{ m \in Z:A^{m^{2}}+A^{m}= 3I -A^{- 6} \right\}, where A=[2−110]A =\begin{bmatrix} 2 & - 1 \\ 1 & 0 \end{bmatrix}. Then n(S)n(S) is equal to

  7. Question 7

    21 Jan 2026 S2

    64
    For the matrices A=[3−41−1]A =\begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix} and B=[−2949−1318]B =\begin{bmatrix} - 29 & 49 \\ - 13 & 18 \end{bmatrix}, if (A15+B)[xy]=[00]\left( A^{15}+ B \right)\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \end{bmatrix}, then among the following which one is true?

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  8. Question 8

    28 Jan 2026 S2

    75
    Let A=[3−41−1]A = \begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix} and BB be two matrices such that A100=100B+IA^{100} = 100B + I. Then the sum of all the elements of B100B^{100} is ____\_\_\_\_ .

  9. Question 9

    6 Apr 2026 S2

    55
    Let A=[100310931]A =\begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix} and B=[bij],1≤i,j≤3B =\left\lbrack b_{ij} \right\rbrack,1 \leq i,j \leq 3. If B=A99−IB=A^{99}-I, then the value of b31−b21b32\frac{b_{31}-b_{21}}{b_{32}} is :

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