How Do You Prove Matrix Powers Using Mathematical Induction?

In summary, the conversation is about using mathematical induction to prove a matrix equation involving A = (2 0 0 3)^n. The person is struggling with understanding the problem and has tried looking for explanations online. Another person suggests trying a couple of examples and using matrix multiplication to solve it. The conversation also mentions the forum's regulations and the importance of showing your own attempt at solving the problem.
  • #1
rakileh
3
0
Hey guys I am in precalculus right now and we just started picking up mathematical induction. Our teacher assigned us a problem that I am stumped over and I tried looking all over for a clear explanation online but I can't find anything remotely helpful. The question is:

Use mathematical induction to prove A = (2 0
0 3)^n = (2^n 0
0 3^n) for every positive integer n

Thank you!
 
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  • #2
These are matrices, correct? Why not just try a couple examples of n, say n = 1. Then n = 2. Then n = 3. The pattern should be clear then and you could make a more general proof.

Matrix multiplication is all you need, e.g.: [A]^2 = [A]*[A]
 
  • #3
Did you read the regulations that you agreed to abide by when you registered for this forum? You must make a valid attempt to solve the problem yourself and show your attempt here. Do you know how to multiply matrices? Do you know what "proof by induction" is?
 

Related to How Do You Prove Matrix Powers Using Mathematical Induction?

1. What is mathematical induction?

Mathematical induction is a proof technique used to prove that a statement is true for all natural numbers. It involves proving that the statement is true for the first natural number, and then showing that if it is true for any arbitrary natural number, it must also be true for the next natural number.

2. When should I use mathematical induction?

Mathematical induction should be used when you need to prove that a statement is true for all natural numbers. It is especially useful for proving theorems and formulas that involve natural numbers.

3. How do I write a proof using mathematical induction?

To write a proof using mathematical induction, you need to follow three steps: 1) State the base case, which is when the statement is true for the first natural number. 2) Assume that the statement is true for an arbitrary natural number, and use this assumption to prove that it is also true for the next natural number. 3) Conclude that the statement is true for all natural numbers by using the principle of mathematical induction.

4. What are some common mistakes to avoid when using mathematical induction?

Some common mistakes to avoid when using mathematical induction include: not clearly stating the base case, assuming that the statement is true for all natural numbers instead of just the next natural number, and using circular reasoning.

5. Can mathematical induction be used for statements that involve real numbers?

No, mathematical induction can only be used for statements that involve natural numbers. This is because the principle of mathematical induction is based on the well-ordering property of natural numbers, which states that every non-empty subset of natural numbers has a smallest element.

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