4. Let A and B be 4 x 4 matrices. Suppose det A = 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det (A?)? (d) (4 points) What is det(A-?)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. and 2 6. (6 points) Let...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a) Find the eigenvalues and eigenvectors of A. (b) Find the algebraic multiplicity an, and the geometric multiplicity, g, of each eigenvalue. (c) For one of the eigenvalues you should have gi < az. (If not, redo the preceding parts!) Find a generalized eigenvector for this eigenvalue. (d) Verify that the eigenvectors and generalized eigenvectors are all linearly independent. (e) Find a fundamental set of...
4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-1)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. [2] and us 6. (6 points) Let vi...
linear algebra 1 0 0 0 1 1 0 0 1. Let A be the 4 x 4 matrix: A= 1 1 1 0 1 1 1 1 (a) Find A-1 by hand. (b) Let T be defined by T(T) = Aē. Use your answer to (a) to find all vectors T E R4 such that: T(T) = 2 4 -9 0 (c) Which of the following statements are true? (Select all that apply) A. The columns of A form...
Matrix Methods/Linear Algebra: Please show all work and justify the answer! 4. Let A and B be 4 x 4 matrices. Suppose det A = 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(A”)? (d) (4 points) What is det(A-")? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of...
4. Let A and B be 4 x 4 matrices. Suppose det A = 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-')? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of A™ are linearly independent. and t = [ ] 6. (6...
Select True or False. No work is required. Let A= [o 1 2 0 4 and y = [6 3]. lil 1. True or False: The Eigenvalues of A are -1 and 4. 3 2. True or False: is an Eigenvector of A. 1 3. True or False: The columns of A are linearly independent. 4. True or False: The columns of A form a basis for R2. 5. True or False: The rank of A is 3. 6. True...
Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent. Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent.
Let A = 4 0 0 2 1 2 1 2 1 (a)(4 marks) Find the eigenvalues of A. (b)(2 marks) Explain without any more calculations that A is diagonalisable. ((7 marks) Find three linearly independent eigenvectors of the matrix A. (d)(2 marks) Write an invertible matrix P such that -100 P-AP=0 40 0 0 3
4 (1) Find a matrix A „such that (A - 41)-1 3 1 (2) Let A be 3x3 matrix with 4 = 4 Find : (a) det(( 3 A)?(2 A)-') (b) det( 2 A-' + 3 adj (A)) (3)Find the values of a that makes the system has (a) unique solution (b) No Solution. 3 A 7 (4)Find the rank of a matrix 17 0 1 2 (5)Suppose that I : R3 → R2 „such that 2 T (e.) =...