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Induction is a powerful mathematical technique used to prove statements for all natural numbers. It works in two main steps: the base case and the inductive step. - **Base Case**: This is where we verify that the statement holds true for the initial value, commonly when the variable is the smallest integer under consideration.- **Inductive Step**: In this step, we assume the statement holds for some specific integer, often denoted as \( k \). Based on this assumption, we prove that the statement also holds for \( k+1 \).In our exercise, induction helps us verify the pattern of matrix powers. By checking that the base case \( A^2 = 2A - I \) is true, we can assume some pattern holds for matrix power \( A^k \). Then, we use this assumption to show that if it is true for \( A^k \), it is also true for \( A^{k+1} \). This completes the induction process, confirming our hypothesis for all \( n \geq 2 \).

The identity matrix, denoted as \( I \), is a special type of matrix with 1s on the diagonal and 0s elsewhere. It acts as the multiplicative identity in the realm of matrices, much like the number 1 in arithmetic. - When any matrix \( A \) is multiplied by an identity matrix \( I \) of the compatible size, the result is the matrix \( A \) itself. This property is crucial in equations like our given \( A^2 = 2A - I \).- Additionally, since \( I \) maintains the structure of the matrix multiplication, it simplifies many processes in proofs and calculations, such as establishing patterns that involve matrices, like recursive relations.The identity matrix serves as a reference point in our solution, helping define when and how matrices transform. This utility is reflected in expressing matrix powers that reuse patterns involving \( I \).

Recursive relations describe sequences where each term is defined as a function of preceding terms. Such relations are invaluable in mathematics for defining sequences based on simpler calculations. - A recursive relation breaks down a complex problem by reducing it to simpler subproblems.- In our context, the recursive relation originates from the given equation \( A^2 = 2A - I \). This equation sets up how we can compute subsequent powers of matrix \( A \).For the third step in our exercise, knowing that \( A^k = 2^{k-1}A - (k-1)I \) sets up a base to effectively anticipate the next power \( A^{k+1} \). By substituting this relation into subsequent calculations, we efficiently build up the chain of matrix powers, utilizing the recursive logic to handle transitions from one power of \( A \) to the next. This logic helps in identifying patterns in matrix exponentiation and validates our final solution, matching option (C).