Question 1:Let A be a non-square matrix, say m x n for m is different from n, Then C = $\displaystyle AA^t$ and $\displaystyle D = A^tA $ are both matrices, but of different sizes.

Show that C and D both have the same non-zero eigenvalues. [Hint: write down the equation that must be satisfied by a vector v to be an eigenvalue $\displaystyle \lambda$, and try to adjust that equation so as to find an eigenvector of D with the same eigenvalue. you will need to explain why your proposed eigenvector is not O.]

Question 2:Is there a 3 x 3 symmetric matrix with eigenvalues $\displaystyle \lambda_{1} = 3, \lambda_{2} = 7, \lambda_{3} = 1 $ and corresponding eigenvectors $\displaystyle v_{1} = (1,0,0), v_{2} = (0,1,1), v_{3} = (0,1,-1)? $ Either find the matrix or explain why it can not exist.