(Matrix Algebra: Diagonalization). Consider the matrix A 3 4 = (4 -9) -6 Verify whether or not A is diagonalizable, and if yes, find a matrix X which diagonalizes it. (i) Start with finding the characteristic polynomial p(2) of A (please type lambda for 1; please DO NOT enter the Unicode character(s) for 1, your input(s) will be ignored otherwise): p() = (ii) Find the eigenvalues of A, and enter them in the increasing order separated by comma(s) into the input field below (if A has no eigenvalues, enter none in the input field): Answer: If you claim that the matrix A has no eigenvalues, skip the remainder of the question. (iii) (a) For each of the eigenvalues of A, write down and obtain the general solution of the homogeneous linear system AD ( x ) = (8) (A - I) X2 whose solution set is equal to the eigenspace; obtain then the fundamental set of solutions (FSS) of the above system, thereby getting a basis for the eigenspace V; in order to simplify the further steps, it is recommended to multiply the vectors of the FSS by suitable integer scalars to change their coordinates to integer numbers. (b) Show your work by providing the following inputs. let be the minimal eigenvalue of A; then the coordinates of the vectors c; of a basis for the eigenspace V you have found in (a) are as follows (enter coordinates of the vectors of the basis one-by-one separated by commas; enter \texttt {unnecessary} in the second input field if your basis has less than two vectors): accordingly, dim(V) = move to the next part (iv), if the matrix A has exactly one eigenvalue; if not, provide the coordinates of vectors d that form a basis for the second eigenspace V (proceed as you did before when entering the coordinates of the vectors Ci): d = = accordingly, dim(Viz = (iv) Now, by (iii), the matrix A is is not diagonalizable. If you claim that A is not diagonalizable, skip the remaining steps. (v) Find a matrix X which diagonalizes A (enter your matrix row-by-row in the two input fields provided; the entries in each row are to be separated by single spaces): (vi) Find the inverse X- of the matrix X from (v), and enter it row-by-row in the input fields below: (vii) Finally, verify your results by finding the matrix and entering it in the input fields below: X-AX