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

**Fourier** · September 6th 2007, 01:03 PM

Hello,

I am stuck on the following problem:

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

Here is what I have started:

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

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

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

**topsquark** · September 6th 2007, 01:21 PM

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
$(cA)^p = c^p A^p$

Here is what I have started:

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

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

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

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

-Dan

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