2022 SQA Advanced Higher Mathematics Paper 2 No.9 Proof by Induction

Question involving the nth power of a matrix.
Instructional exercise consisting of question 9 from Paper 2 of the 2022 SQA Advanced Higher Mathematics examination.

Where appropriate the alternative methods for solving the problem are also given.
Set out in stages of 'try then check' the worked solution to each part of the question.
Including indications of where each mark is allocated, and the requirements for that mark, according to the marking scheme.