NAME:Math 0280 - Intro Matrices and Linear Algebra Midterm 2ACIRCLE ONE: Grade my [WORK & ANSWERS] or [ANSWERS ONLY]Answer each of the following. If you wish to receive partial credit, please show all work (No Work =No Credit.) In particular, credit will not be given for a “correct answer” when the accompanying work isnonexistent or silly (unless you have selected “Grade my Answers Only”). Academic misconduct will resultin an exam score of zero, possible expulsion from the class, and possible dismissal from the University.1. (a) Using Gauss-Jordan elimination, find the inverse or explain why it does not exist for the followingmatrices:1. (6 points) A =1 0 30 1 21 0 42. (6 points) B =1 1 0 00 1 1 01 0 0 10 0 1 1(b) (10 points) Give a basis for row(B), col(B), and null(B) where B is the matrix in part (a2).(c) (4 points) Do the column vectors of A in part (a1) form a basis for R3? To receive credit you mustjustify your answer.(d) (6 points) Write the matrix A in part (a1) as a product of elementary matrices.Page 22. (6 points) Solve the given matrix equation for X and simplify your answer as much as possible. Assumeeach matrix is invertible and each matrix multiplication is defined.BXA = (B−1A)23. (9 points) Prove that the following transformation is a linear transformation:Txy=x + yx − y.Page 34. For the linear transformationsSxy=2xx + y.andTxy=2x − 5y3y.(a) (6 points) Find [S] and [T ].(b) (5 points) Find S ◦ T .(c) (6 points) Is S invertible? Justify your answer. If S is invertible, find [S−1].Page 45. (10 points) Let P : R2→ R2be the projection onto the line y = −3x. Given that this is a lineartransformation, find [P ].Page 56. (8 points) Show λ = 4 is an eigenvalue for A =1 23 2and find one eigenvector.Page 67. (10 points) Evaluate the following determinants:(a) (5 points)0 3 0 0 00 0 0 5 0−2 0 0 0 00 0 1 0 00 0 0 0 2(b) (5 points)1 2 3 4 56 −5 4 −3 20 0 1 0 00 −1 2 −5 31 2 3 4 5Page 7(c) (8 points)2 4 4 −22 5 7 51 2 1 −32 4 4 1Page
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