# Help in proving (cA)^p = c^p*A^p for any matrix A

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• Sep 6th 2007, 01:03 PM
Fourier
Help in proving (cA)^p = c^p*A^p for any matrix A
Hello,

I am stuck on the following problem:

If p is a nonnegative integer and c is a scalar, show that
$\displaystyle (cA)^p = c^p A^p$

Here is what I have started:

$\displaystyle \textrm{Let } A=[a_{ij}] \textrm{ be any } n \times n$matrix. Then

$\displaystyle (cA)^p=([ca_{ij}])^p$

Where could I go from here? Is this a good first start?
• Sep 6th 2007, 01:21 PM
topsquark
Quote:

Originally Posted by Fourier
Hello,

I am stuck on the following problem:

If p is a nonnegative integer and c is a scalar, show that
$\displaystyle (cA)^p = c^p A^p$

Here is what I have started:

$\displaystyle \textrm{Let } A=[a_{ij}] \textrm{ be any } n \times n$matrix. Then

$\displaystyle (cA)^p=([ca_{ij}])^p$

Where could I go from here? Is this a good first start?

Almost there!
$\displaystyle (cA)^p=([ca_{ij}])^p = c^p [a_{ij} ]^p = c^pA^p$

-Dan