5. Suppose A is an n xn invertible matrix such that AT = A. Prove that det (A) = ±1.
19. Suppose A and B are n xn matrices. a. Suppose that both A and B are diagonalizable and that they have the same eigen- vectors. Prove that AB = BA. b. Suppose A has n distinct eigenvalues and AB = BA. Prove that every eigen vector of A is also an eigen vector of B. Conclude that B is diagonalizable. (Query: Need every eigenvector of B be an eigenvector of A?)
9. Let c be a scalar and A an n xn matrix. (a) Suppose that the sum of the entries of each row of A is equal to c. Show that c is an eigenvalue of A. (b) Suppose that the sum of the entries of each column of A is equal to c. Show that c is an eigenvalue of A Hint: Consider the possible eigenvectors associated to the scalar c.
4. RWI 4.5.13). Suppose that the sequence (xn) satisfies n 1,2,.. X2ax.1 + bx and that 0< r<R satisfy - az- b (z-rz- R) (a) Show that x O(R" ). O(2") is false. (b) Give an example with r R 2 for which x, (c) What asymptotic (big-oh) estimate for (x) can you give in general if r R> 0?
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b) If AT = A and if vectors u and v satisfy Au = 3u and Av = 40, then u: v=0. (c) An n x n symmetric matrix has n distinct real eigenvalues. (d) For a nonzero v in R", the matrix vv7 is a rank-1 matrix.
Suppose A is a 3 by 3 matrix. Decide if the matrix is diagonalizable given the following information: A has two distinct eigenvalues 11, 12 whose eigenspaces are a line and a plane, respectively. Not diagonalizable Not enough information O Diagonalizable Question 14 6 pts READ FIRST: Fill in the blanks. ADDITIONALLY, on you scanned work, show how you arrive at your answers. (Your answer must match your work or you will receive no credit.) The set S= {(1,-1,3), (-3,4,9),...
2 is the question
Question 4 [35 marks in total] An n xn matrix A is called a stochastic matriz if it satisfies two conditions: (i) all entries of A are non-negative; and (ii) the sum of entries in each column is one. If the (i, j) entry of A is denoted by aij for i,j e {1, 2, ..., n}, then A is a stochastic matrix when aij > 0 for all i and j and in dij =...
step by step please
Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.) Sufficient Condition for Diagonalization If an n x matrix A has n distinct eigenvalues, then the corresponding cigenvectors are linearly independent and A is diagonalizable 02 Find the eigenvalues. (Enter your answers as a...
Let A = CD where C, D are n xn matrices, and is invertible. Prove that DC is similar to A. Hint: Use Theorem 6.13, and understand that you can choose P and P-inverse. Prove that if A is diagonalizable with n real eigenvalues 11, 12,..., An, then det(A) = 11. Ay n Prove that if A is an orthogonal matrix, then so are A and A'.
6. Suppose Xn is a two-state Markov chain with transition probabilities (Xn, Xn+1), n = 0, 1, 2, Write down the state space of the Markov chain Zo, Zi, . . . and determine the transition probability matrix.