I have a couple of Linear Algebra questions.
1.) Given a Matrix:
A = ([[5/6,1/3,0,-1/6],[1/3,11/42,3/14,4/21],[0,3/14,5/14,3/7],[-1/6,4/21,3/7,23/42]])
Where the numbers in each of the 's is a row (the above matrix corresponding to a 4x4 matrix);
a.) Is the Matrix A diagonalizable? If it is, diagonlize A. If it is not, explain why it is not. Is A invertible? If it is, find A^(-1). If it is not, explain why it is not.
b.) By looking at the above example, we are easily able to cehck that A^(2) = A, and that A^(T) = A, even though A =/ (does not equal) I. Consider the general case of an n x n matrix, B, with B =/ I and B^(2) = B. Show that B is not invertible.
2.) Suppose Matrix A = ([[c_1, c_2, ..., c_n],[c_1, c_2, ..., c_n],[c_1, c_2, ..., c_n],[c_1, c_2, ..., c_n],[c_1, c_2, ..., c_n],[c_1, c_2, ..., c_n]]). Also, assume that the rows of Matrix A do not add up to 0. That is,
Summation(c_i =/ 0) from i=1...n.
Find all of the eigenvalues, their (algebraic) multiplicities, and as well as the dimensions of their eigenspaces.