Number Theory Matrix Help

• March 6th 2009, 10:08 AM
tokio
Number Theory Matrix Help
Suppose that a matrix A =
a 0

0 b

where a and b are real numbers

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

for every positive integer n
• March 6th 2009, 11:09 AM
Matrix proof
Hello tokio
Quote:

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

0 b

where a and b are real numbers

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

for every positive integer n

Use induction: it takes one line:

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

I'll leave the rest to you.

• March 6th 2009, 06:33 PM
tokio
Quote:

Hello tokioUse induction: it takes one line:

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

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.
• March 6th 2009, 10:33 PM
Given that $A = \begin{pmatrix}a & 0\\0 & b\end{pmatrix}$, let $P(n)$ be the propositional function: $A^n =\begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}$
Then $P(n) \Rightarrow A^{n+1}= \begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}\begin{pmatrix}a & 0\\0 & b\end{pmatrix} = \begin{pmatrix}a^{n+1} & 0\\0 & b^{n+1}\end{pmatrix}\Rightarrow P(n+1)$
$P(1)$ is true. So $A^n =\begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}, \forall n \in \mathbb{N}$