for this markov matrix [9/10 3/5 0] [1/10 3/10 1/2] [0 1/10 1/2] how could i show that the eigenvectors v1 = [30 5 1]¹ V2 = [−2, 1, 1]¹, and v3 = [3, −4, 1]¹ are linearly independent by confirming c1 = c2 = c3 = 0 is a unique solution to the sum from Σ-3(C₁) (V₂) = 0 in terms of a linear combination of a eigenvectors, Find the general solution to the stochastic process at the nth iteration

We have the following matrix where each column is identical: a а а А — b 1. Since this matrix is not full rank, it is singular and thus 1 = 0 is an eigenvalue. Find two linearly independent eigenvectors of A corresponding to 1 = 0. You may use SymPy or other libraries to check your work, but please show the steps of your would-be computations. 2. 1 = 0 is also an eigenvalue of A". Find two linearly independent eigenvectors of AT corresponding to 1 = 0. 3. A and A' have three eigenvalues each, but 1 = 0 makes up two of the three since it corresponds to two linearly independent eigenvectors for both matrices. Find the third eigenvalue of A and A' by looking at the trace of A. 4. Show that (a, b, c) is the corresponding eigenvector of A and that (1, 1, 1) is the corresponding eigenvector of A' .

Suppose that A is an n×n matrix with n eigenvalues μ1, μ2, ..., μn. Show that the determinant of A is equal to the product of its eigenvlaues.

If matrix A is square of size n x n with a rank equal to 2 and n > 2. The rank of a matrix is equal to the number of non-zero eigenvalues. Can you write A as the product of two matrices B E Rnxq and C E RPX" where we require that p < n and q < n? You may assume that A is diagonalizable. Show how to construct a matrix B and C such that A = BC.