1. Number Theory Matrix Help

Suppose that a matrix A =
a 0

0 b

where a and b are real numbers

Prove that $\displaystyle A^n$=
$\displaystyle a^n$ 0
0 $\displaystyle b^n$

for every positive integer n

2. Matrix proof

Hello tokio
Originally Posted by tokio
Suppose that a matrix A =
a 0

0 b

where a and b are real numbers

Prove that $\displaystyle A^n$=
$\displaystyle a^n$ 0
0 $\displaystyle b^n$

for every positive integer n
Use induction: it takes one line:

$\displaystyle \begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}\begin{pmatrix}a & 0\\0 & b\end{pmatrix} = \dots$

I'll leave the rest to you.

Hello tokioUse induction: it takes one line:

$\displaystyle \begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}\begin{pmatrix}a & 0\\0 & b\end{pmatrix} = \dots$

I'll leave the rest to you.

I'm confused on how to set this up to do the induction , im trying the hardest to understand, things are just not clicking right now.

4. Proof by induction

Hello Tokio

Given that $\displaystyle A = \begin{pmatrix}a & 0\\0 & b\end{pmatrix}$, let $\displaystyle P(n)$ be the propositional function: $\displaystyle A^n =\begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}$

Then $\displaystyle P(n) \Rightarrow A^{n+1}= \begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}\begin{pmatrix}a & 0\\0 & b\end{pmatrix} = \begin{pmatrix}a^{n+1} & 0\\0 & b^{n+1}\end{pmatrix}\Rightarrow P(n+1)$

$\displaystyle P(1)$ is true. So $\displaystyle A^n =\begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}, \forall n \in \mathbb{N}$