Name: Due Date: Tues. Dec. 5th In Class Peer Review Assignment 2 D.M. 1 : 7 (7pts) Short Answer 8 : 14 (32pts) T/F and Multiple Choice 15 : 30 (15pts) Total out of (54pts) Directions: Put only your answers on this assignment. Do all scratch work and side calculation on a separate piece of scratch paper. If you need to attach work please staple it on the back of this assignment. Write neatly and legibly. If I can t read it you will receive a zero for that problem. Finally be sure to show all of your work (justification) for each problem to receive full credit. Grader Names: 1
Table 1: Some key terms from exam 2 material. Singular Non-Singular Invertible Inverse Determinant Cofactor Cofactor Expansion Adjoint Subspace Column Space Null Space Basis Dimension Coordinate Vector Rank Nullity Eigenvector Eigenvalue Eigenspace Cramers Rule Definition Matching: (1pt each) Choose the most appropriate key term in Table 1. 1. A matrix that is not invertible is known as a matrix. 2. Given A R n m the set of all solutions to the homogeneous equation Ax = 0 is known as the of A. 3. Given A R n n if M R n n s.t. AM = MA = I then M is defined to be the of A. 4. Given a subspace W, if S is linearly independent set of vectors whose span is W then S is a of W. 5. If A R n m, x 0 and c R s.t. Ax = cx then c is defined to be a of A. 6. Given a subspace W, the number of vectors in a basis of W is defined to be the of W. 7. Given A R n m, the number of vectors in a basis of Col(A) is known as the of A. 2
Proof and Short Answer: [ ] [ ] a b 8. (2pts) Let A = prove that if ad bc 0 then A c d 1 1 d b = ad bc c a 9. (2pts) Prove that if A 1, A 2,..., A n are all invertible m m matrices then so is their product. 3
10. Let A R m n. (a) (2pts) Prove that Null(A) is a subspace. (b) (2pts) Prove that if λ is an eigenvalue of A then the eigenspace of λ is a subspace. 4
11. Let A R 4 4. (a) (1pt) Write the cofactor expansion definition of the determinant of a m n matrix. Be sure to define what the cofactor C ij represents. (b) (2pts) Use the cofactor expansion definition to find the determinant of 5 7 2 2 A = 0 3 0 4 5 8 0 3 0 5 0 6 along the fourth row. (c) (2pts) Verify your work in part (b) by showing you get the same result when expanding along the third column. 5
12. Consider the system x 1 + 3x 2 + x 3 = 4 x 1 + 2x 3 = 2 3x 1 + x 2 = 2 (a) (2pts) Use Cramers rule to solve this system. 6
(b) (1pt) State the inverse matrix formula. Be sure to define all terms. (c) (2pts) Use the inverse formula to find the inverse to the coefficient matrix of the system and show your solution in (b) is valid by computing A 1 4 2. 2 7
(d) (2pts) Find the volume of the parallelepiped determined by the columns of the system. 13. (2pts) Prove that 0 is an eigenvalue of a m n matrix, A, rank(a) < n. 8
1 2 1 1 14. Let u = 2 and A = 0 1 0 2 1 1 2 (a) (2pts) is u Col(A)? (b) (2pts) What is the rank of A? (c) (2pts) What is the nullity of A? (d) (2pts) Find two different basis s for the Col(A). (e) (2pts) Find two different basis s for the Null(A). 9
T/F and Multiple Choice: (1pt each) 15. : The row operation r i + cr j r j preserves the value of the determinant of a matrix for any c R. 16. : Row swapping changes the sign of the determinant. 17. : det ( A T A ) 0 18. : Scaling a row by some c R \ {0} also scales the determinant by c. 19. : The determinant of a 2 2 matrix, A, represents the area of the paralelogram formed by the columns of A. 20. : If A is an invertible n n matrix then = det(b)det(c) if A = BC. 21. : If A R n n and λ = 0 is an eigenvalue of A, then the transformation T(x) = Ax is one-to-one. 22. : A matrix A is not invertible if and only if 0 is an eigenvalue of A. 23. : A number c is an eigenvalue of A if and only if (A ci)x = 0 has nontrivial solutions. 24. : The eigenvalues of a matrix are on its main diagonal. 25. : If the columns of a matrix A R n n are linearly dependent, then = 0. 26. If A R n n then det(a T ) = : (a) 1, (b) 1, (c), (d) 27. If A R n n then det(a 1 ) = : (a) 1, (b) 1, (c), (d) 28. If A R n n and B R n n then det(ab) = : (a) +det(b), (b) det(b), (c) det(b), (d) det(b), (e) det(b) 29. If A R n n, m N then det(a m ) = : (a) m, (b) m, (c) m, (d) m 30. If S R 2 and has [ an area ] of 12 square units and T is a linear transformation with standard transformation 1 2 matrix given by then the area of T(S) is : 3 2 (a) 48, (b) 48, (c) 3, (d) 3 10