Question 4 [10 points] Find a formula in terms of k for the entries of A",...
Find a formula in terms of k for the entries of A, where A is the diagonalizable matrix below and P TAP-D for the matrices P and D below. 00
Question 3 [10 points] Find a formula in terms of k for the entries of Ak, where A is the Hermitian matrix below A= Ak= -19382) (-3)*- * (3)+(3+21) ${3}K={{2}% (-3)*(+5–53– 4 (2) The correct answer is: Ak 16-12)+(36)-3)* 12+)+-3)*
-8 -24 -12 (16 points) Let A= 0 4 0 6 12 10 (a) (4 points) Find the eigenvalues of A. (b) [6 points) For each eigenvalue of A, find a basis for the eigenspace of (b) [6 points) is the matrix A diagonalizable? If so, find matrices D and P such that is a diagonal matrix and A = PDP 1. If not, explain carefully why not.
1 2 2 1 -X Find the determinant of the matrix as a formula in terms of x and y. Remember to use the correct syntax for a formula 0 0 1 -3 -x X Question 4: (2 points) a b c fis 3, find the determinant of these matrices: If the determinant of the matrix M = d e (gh k) b a C (a) 7 d 7e 7f h k -E. b-2 e c - 2 f a...
Question 7 [10 points) Find conditions on k that will make the matrix A invertible. To enter your answer first select 'always', never', or whether should be equal or not equal to specific values, then enter a value or a list of values separated by commas. k 0 2 A-8k-2 4 2-1 A is invertible: Always Always Never Question 8 [10 When k = Express the follow. When katrix A as a product of elementary matrices:
Part 4 of 10 - Question 4 of 10 1.0 Points Find k such that Pr[Z<k] = .7517, where Z is the standard normal random variable. O A..2483 O B.-.32 Oc..32 O D.-.68 O E..68 Reset Selection
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP^−1. If not, explain carefully why not.
Let matrix M = -8 -24 12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP−1. If not, explain carefully why not.
Q3. Given the following matrices, A=[ 3), B =[10], C= [31 a. Find the characteristic polynomial of A, B, C respectively. b. Is A diagonalizable? Is B diagonalizable? Is C diagonalizable? If no, please state your reason. If yes, please find the matrix P and D such that p-1MP = D c. Is the matrix A similar to the matrix C? Please explain your answer briefly.
5. Let (a) (2 marks) Find all eigenvalues of A (b) (4 marks) Find an orthonormal basis for each eigenspace of A (you may find an orthonormal basis by inspection or use the Gram-Schmidt algorithm on each eigenspace) (c) (2 marks) Deduce that A is orthogonally diagonalizable. Write down an orthogonal matrix P and a diagonal matrix D such that D P-AP. (d) (1 mark) Use the fact that P is an orthogonal matrix to find P-1 (e) (2 marks)...