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Math 240 Fall 2013 Study problems for Exam1 continued Do not use a calculator. 7. Match the slope fields with their differential equations. (A) (C) (D) dy x dx 1 0 1 2 3 # 8. Assume we are given Ax=b with augmented matrix A = 0 1 0 0 5 0 0 0 0 0 # a) Is A in RREF? If not, use elementary row operations to find a row equivalent matrix that is. b) what is the rank of A? c) What is the rank of A#?. d) Does the augmented matrix represent a consistent system? e) How many variables are in the system? f) are there any free variables, if so how many? g) Write the solution set. 1 0 2 0 9) Use elementary row operations to find the RREF of A# = 0 1 1 7 then write the 0 0 2 4 1. dy sin x dx (B) 2. dy x y dx 3. dy 2 y dx 4. solution to the corresponding system . 10. If A and B are both n x n matrices, is the following correct? (If not, state the condition(s) which makes it correct.) (A + B)2= A2 - 2AB + B2 2 3 11. Given A, find the most general matrix B such that AB= 0. A = 0 0 12. Find a matrix B (or state it is impossible) such that AB = AT. (Use A from problem # 11) 13. Let A = diag(1,2,3,w) where w is an arbitrary real number. what is A5? What is A-1? Explain. 14. A matrix A is called involutory if A2= I. Carefully prove that if A is involutory then (I – A) (I + A) = 0 Math 240 Fall 2013 Study problems for Exam1 continued Some solutions (not guaranteed to be correct!) 3s / 2 3t / 2 # 8g {(3+2s-t, 5, s, t)} #9 {(5,5,2)} #10 no # 11 s, t arbitrary real t s numbers. 12. Impossible. 13. Diag (1,32,243, w5)