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...
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...
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 points Suppose A, B, and Care 5 x 5 matrices with det(A) = -2, det(B) = 10 and the columns of C are linearly dependent. Find the following or state that there is not enough information: (a) det(10B-) (b) det(AB) (c) det(CA+CB)
11. Prove one of the following: a. Let A and B be square matrices. If det(AB) + 0, explain why B is invertible. b. Suppose A is an nxn matrix and the equation Ax = 0 has a nontrivial solution. Explain why Rank A<n.
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix, 5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent. Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB 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 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.
1. (10 points) Let A and B be 3 x 3 matrices, with det A = -3 and det B = 2. Compute (a) det AB (6) det B4 (c) det 3B (d) det A"B" AT (e) det B-AB
SOLVE BOTH 4 and 5!! 4. Let A and B be two nxn matrices. Suppose that AB is invertible. Show that the system Ar 0 has only the trivial solution 5. Given that B and D are invertible matrices of orders n and p respectively, and A = Find A by writing A as a suitably partitioned matrix