2. (8 points) Let B be the matrix given to the right. Calculate det(B) by expanding through the second row and again by expanding through the third column. (All work must be shown by hand.) 5 5 4 -3 -3 60
(10 pts) If the determinant of a 5 x 5 matrix A is det (A) = 8, and the matrix B is obtained from A by multiplying the second column by 9, then det (B) =
Prove the following lemma. Let B be an n ✕ n matrix and let E be an n ✕ n elementary matrix. Then det(EB) = det(E) det(B) 1. Write the proof and submit as a free response. (Submit a file with a maximum size of 1 MB.) 2. Which of the following could begin a direct proof of the statement? If E interchanges two rows, then det(E) = 1 by Theorem 4.4. Also, EB is the same as B but...
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...
Please answer this using matrices quick thanks 1. Let A be a 3 x 3 matrix with det (A) 4, and suppose the matrix B is obtained from A by performing the following elementary row/column operations to A: -a Ra+ Rs For what value(s) of a does det(B)-6?
2 invertible? C For which values of c is the matrix 8 O c 4 c =-4 Both of the above, i.e., c +4 Neither of the above, i.e., c +4. Suppose that the following row operations: interchange rows 1 and 3 multiply row 3 by 1/2 add -3 times row 1 to row 2 2 1 7 in this order, transform a matrix A into B = | 0 4-5 L0 0 3 What is the determinant of A?...
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...
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...
Problem 2. (50 points) Suppose that a 4 x 4 matrix A with rows it. 73, 74, and has determinant det A = 1. Find the following determinants: ☺ U2 ü det 604 det det A +50 Note: You can earn partial creat on this problem Problem 1. (50 points) Let Rem -5 -20 -1 -1 - 1 4 16 0 الها (a) Compute dexA Use Cramer's rule to solve the following system 20x -5x -X -4 3 + 16x9...
a. The elementary matrix ?1E1 multiplies the first row of A by 1/6.b. The elementary matrix ?2E2 multiplies the second row of A by -4.c. The elementary matrix ?3E3 switches the first and second rows of A.d. The elementary matrix ?4E4 adds 6 times the first row of A to the second row of A.e. The elementary matrix ?5E5 multiplies the second row of B by 1/3.f. The elementary matrix ?6E6 multiplies the third row of B by -4.g. The elementary matrix ?7E7 switches the first and third rows of B.h. The elementary matrix ?8E8 adds...