Let A be a diagonalizable n x n matrix and let P be an invertible n...
Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = −4 0 4 −3 −1 4 −6 0 6 , A5
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that Ak = pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. -10 -18 A = 6 11 18].45 -253 -378 A6 = 126 188 11
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that A* = Pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. 10 18 A = -6 -11 18].46 A = 11
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det (A) > 0. (1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
(1 point) Let A= [44 18 (18 -45 -19 –18 -60 -24Find an invertible matrix P and a diagonal matrix D such that D=P-1AP. –25] T ! 0
37 40 -120 1 point) Let 5 -815Find an invertible matrix P and a diagonal matrix D 10 10 -33 such that D P-1AP
1. For a polynomial p(1) = cktk + Ck-14k-1 +...+ci+co, and an n x n matrix A, we define p(A) = CkAk + Ck-1 Ak-1 + ... +CjA + col. Let A be an n x n diagonalizable matrix with characteristic polynomial PA(1) = (1-2)*(1-3)n-k where 1 <k<n - 1. In other words, let A be an n x n diagonal- izable matrix that has only 2 and 3 as eigenvalues. Explain what is wrong with the following false "proof...
please solve both 3. [-12 Points] DETAILS LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 8 -2 A= P= Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP = 1. [0/2 Points] DETAILS PREVIOUS ANSWERS LARLINALG8 7.2.001. Consider the following. -11 40 A= -27 (a) Verify that A is diagonalizable by computing p-1AP. -1 0 p-1AP = 10 3...
29&30 please 3 -23 4 3-2 25. 3 4926. |0 1 1 0 0-2 1 2-5 Finding a Basis In Exercises 27-30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. 27. T: R2→R-T(x, y) = (x + y, x + y) 28. T: R3→R, Tu, y, z) (-2x +2y -3z, 2r y -6z. 2y) a + (af+ 2b)s 29. T: Pi-Pi T(a + bx) 30. T: P㈠Pg Tle...