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)...
a.) if A is an m*n matrix, such that Ax=0 for every vector x in R^n, then A is the m * n Zero matrix b.) The row echelon form of an invertible 3 * 3 matrix is invertible c.) If A is an m*n matrix and the equation Ax=0 has only the trivial solution, then the columns of A are linearly independent. d.) If T is the linear transformation whose standard matrix is an m*n matrix A and the...
For the following problems use: Annx n matrix A is invertible RREF(A) = I rank(A) - n A 2 x 2 matrix A is invertible = det(A) 0 3 singular (non-invertible). For which value(s) of h is A = -2 -1 -4 Choose... Choose... 6 2 h-2 a 0,b 0,c+0,d +0 A = 4 -1 C 0 x-2 or x 4 For which values of x is A = invertible a 0,b 0,c 0,d=0 4 x 2 X#1 and x2...
[-2.00 Consider a 2 x 2 matrix A = | | 0.00 matrix D such that A = PDP-1. 0.00 ] . Find an invertible 2 x 2-matrix P and a diagonal 2 x 2- -2.00 P = Note: In order to be accepted as correct, all entries of the matrix A – PDP-1 must have absolute value smaller than 0.05.
9. An n × n matrix A is called nilpotent if for-one non, negalivew m, we have Ao, If A is a nilpotent matrix prov conider invertible matrix. To prove this tell me what is (1 + AY first the case where m2 and in this case show th This should help you to see how to prove the general n x n identity matrix). that 1+ As an Hin at (1+A)---A) case. (I is the 9. An n ×...
5. Let G be a graph with order n and size m. Suppose that n 2 3 and n-n2)+2 m > Using Ore's Theorem, prove that G is Hamiltonian 5. Let G be a graph with order n and size m. Suppose that n 2 3 and n-n2)+2 m > Using Ore's Theorem, prove that G is Hamiltonian
Explain all parts of question 1 and question 2 in detail 1. Consider the matrix In + Inn, which has every diagonal entry equal to 2 and every off-diagonal entry equal to 1. (a) Compute det(In + Inn) for each of n = 1,2,3. (b) For n = 4, we have 2 1 1 1 1 2 1 1 1 1 2 1 111 2 2 1 1 1 -1 1 0 0 -1 0 1 0 -1 0 0...
(5 points) Consider the following matrix for CER. [ 2 c cl I c 5 c | 8 cc Find all values of c for which the matrix is invertible (including an argument showing when it is invertible or not).
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2. An eigenvector corresponding to the eigenvalue 1 is . Find an invertible matrix M such that M−1AM= ⎛⎝⎜⎜⎜1000-2000-2⎞⎠⎟⎟⎟. Enter the Matrix M in the box below. Question 8: Score 0/2 1 3 -3 4 6 -6 8 The matrix A = 1-6 6 | is diagonalisable with eigenvalues 1,-2 and-2. An eigenvector corresponding to the eigenvalue 1 is -2 2 1 0 0 0 0-2 Find an invertible matrix...
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.