Answer each of the following questions withjustification.(a) If A is a 5 x 6 matrix, can the columns of Aform a basis for R5.(b) If A, B, and C are n × n matrices, A isinvertible, and AB = AC, then must B = C?(c) If x is an eigenvector of the 4 x 4 matrix Acorresponding to the eigenvalue 2 = 0, do thecolumns of A span Rª?(d) If A is a 3 x 4 matrix, what is the largest valueof the rank of A? What is the largest value ofthe dimension of the null space of A?(e) Let A be a 4 x 4 matrix with det A = 6, and thematrix B is formed from A by firstinterchanging Rows two and three, and thendividing Row one by 2. What is det B?%3D

Answer each of the following questions with justification. (a) If A is a 5 x 6 matrix, can the columns of A form a basis for R5. (b) If A, B, and C are n x n matrices, A is invertible, and AB = AC, then must B = C? (c) If x is an eigenvector of the 4 x 4 matrix A corresponding to the eigenvalue 2 = 0, do the columns of A span R*? (d) If A is a 3 x 4 matrix, what is the largest value of the rank of A? What is the largest value of the dimension of the null space of A? (e) Let A be a 4 x 4 matrix with det A = 6, and the matrix B is formed from A by first interchanging Rows two and three, and then dividing Row one by 2. What is det B? %3D

Answer each of the following questions with justification. (a) If A is a 5 x 6 matrix, can the columns of A form a basis for R5. (b) If A, B, and C are n x n matrices, A is invertible, and AB = AC, then must B = C? (c) If x is an eigenvector of the 4 x 4 matrix A corresponding to the eigenvalue 2 = 0, do the columns of A span R*? (d) If A is a 3 x 4 matrix, what is the largest value of the rank of A? What is the largest value of the dimension of the null space of A? (e) Let A be a 4 x 4 matrix with det A = 6, and the matrix B is formed from A by first interchanging Rows two and three, and then dividing Row one by 2. What is det B? %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Consider the following matrix: 3. -2 A =-14 15 18 -18 -10 (a) Find the eigenvalues of A. (b) Find three linearly independent eigenvectors for A and state the algebraic and geometric multiplicity for each eigenvalue. Form a matrix C whose columns are the three linearly independent eigenvectors that you found in (b). Verify that C-1 AC is a diagonal matrix (which we call D). (d) Use your previous answer to find a matrix B with the property that B3 = A. (e) For any real matrix A, write down a sufficient condition for a real matrix E to exist such that E2 = A, and prove that this condition is sufficient. %3D
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