Question

Challenge Problem Use the Principle of Mathematical Induction to prove that $$ \left[\begin{array}{rr} 5 & -8 \\ 2 & -3 \end{array}\right]^{n}=\left[\begin{array}{cr} 4 n+1 & -8 n \\ 2 n & 1-4 n \end{array}\right] $$ for all natural numbers \(n\).

Short answer

Proof by induction

Step-by-step solution

State the Principle of Mathematical Induction

The Principle of Mathematical Induction involves proving two steps. Base Case: Prove the statement for the initial value (usually n = 0 or n = 1). Inductive Step: Show that if the statement holds for an arbitrary natural number k, then it also holds for k + 1.

Base Case

Prove the base case for n = 1. Evaluate the left-hand side of the given equation:

Evaluate Left-Hand Side

Calculate

Verify Base Case

Calculate the right-hand side.

Inductive Hypothesis

Assume the statement is true for some arbitrary natural number k, i.e.,

Inductive Step

Prove the statement for k + 1.

Compute Left-Hand Side for k + 1

Calculate for the next step.

Simplify and Compare

Show that result matches Inductive Hypothesis for n = k + 1.

Key concepts

Inductive Proof

Inductive proof is a powerful method used to prove statements about natural numbers. It involves two main steps: the base case and the inductive step. By verifying these steps, we can conclude that a statement is true for all natural numbers. This method is especially useful because it builds on previous results to establish the truth of a statement for larger values of n.

Base Case

The base case is the initial step in an inductive proof. It demonstrates that the statement we are trying to prove is true for the smallest value of the natural number, usually n = 0 or n = 1. For example, in the exercise above, we need to check if the given matrix equation holds true when n = 1. This provides a starting point for our proof and ensures that the inductive process has a solid foundation.

Inductive Step

The inductive step is the next part of the inductive proof. Here, we assume that the statement we are trying to prove is true for some arbitrary natural number k. We then need to show that this assumption (called the inductive hypothesis) implies that the statement is also true for k + 1. By proving this, we establish that if the statement is true for one natural number, it must be true for the next, thereby proving it for all natural numbers. This step ensures the continuation of the truth from one value to the next, creating a domino effect that validates the entire proof.