What are the properties of matrices and their powers?

[Max Power]

I have a few problems at which I'm at a loss.

The first problem asks to "determine the elementary matrix that will multiply the third row of a 3x3 matrix by 2/3". I'm not even sure what this problem is asking. If A is a 3x3, is this problem asking for a matrix, B, which when you do AB, the third row of A is multplied by 2/3?

The second problem says "Let lambda be an eigenvalue and v an associated eigenvector of matrix A. Prove that v is also an eigenvector for A^2. What is the associated eigenvlue?" I can't find anything about the properties of matrices and their powers?

Might someone be able to offer a nudge or push in the right direction for either of these problems?

Thank you.

 

[daveb]

The second question is asking, if Av= lambdav, with lambda the eigenvlaue and v the associated eigenvector, prove that v is also an eigenvecotr for A^2, or A^2v = deltav, with delta the eigenvalue.

 

[daveb]

As for the first question, it's asking for a matrix B such that the third row of A is 2/3 its original value when you calculate BA.