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Count 2x2 matrices with entries from a set that satisfy a matrix equation

This type of question, asked 3 times: 2021 Paper 23, 26 Jul 2022, 4 Apr 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    87
    The number of elements in the set
    {A=[ab0 d]:a,b,d∈{−1,0,1} and (I−A)3=I−A3},\left\{ A =\begin{bmatrix} a & b \\ 0 & \text{ }d \end{bmatrix}\mathbf{:a,b,d \in \{ - 1,0,1\}}\text{~and~}(I - A)^{3}= I -A^{3} \right\},
    where I is 2×22 \times 2 identity matrix, is:

  2. Question 2

    26 Jul 2022

    180
    The number of matrices A=[abcd]A =\begin{bmatrix} a & b \\ c & d \end{bmatrix}, where a, b,c,d ∈{−1,0,1,2,3,……,10}\mathbf{\in \{ - 1,0,1,2,3,\ldots\ldots,10\}}, such that A=A−1A =A^{- 1}, is

  3. Question 3

    4 Apr 2026 S1

    55
    Let S={A=[abcd]:a,b,c,d∈{0,1,2,3,4}S =\left\{ A =\begin{bmatrix} a & b \\ c & d \end{bmatrix}\mathbf{:a,b,c,d \in \{ 0,1,2,3,4\}} \right. and  A2−4 A+3I=0}\left. \ A^{2}- 4\text{ }A + 3I = 0 \right\} be a set of 2×22 \times 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4 , is:

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