Pitt MATH 0280 - Midterm 2A (8 pages)

NAME Math 0280 Intro Matrices and Linear Algebra CIRCLE ONE Grade my Midterm 2A 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 is nonexistent or silly unless you have selected Grade my Answers Only Academic misconduct will result in 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 following matrices 1 0 3 1 6 points A 0 1 2 1 0 4 2 6 points 1 0 B 1 0 1 1 0 0 0 1 0 1 0 0 1 1 b c d 10 points Give a basis for row B col B and null B where B is the matrix in part a2 3 4 points Do the column vectors of A in part a1 form a basis for R To receive credit you must justify your answer 6 points Write the matrix A in part a1 as a product of elementary matrices Page 2 2 Solve the given matrix equation for X and simplify your answer as much as possible Assume each matrix is invertible and each matrix multiplication is defined 6 points BXA B 1 A 2 3 9 points Prove that the following transformation is a linear transformation x x y T y x y Page 3 4 For the linear transformations S and T x y x y 2x x y 2x 5y 3y 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 4 5 Let P R2 R2 be the projection onto the line y 3x Given that this is a linear transformation find P 10 points Page 5 6 8 points Show 4 is an eigenvalue for A 1 3 2 2 Page 6 and find one eigenvector 7 10 points a b Evaluate the following determinants 5 points 0 0 2 0 0 5 points 1 6 0 0 1 3 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 2 2 5 0 1 2 3 4 4 3 1 0 2 5 3 4 5 2 0 3 5 Page 7 c 8 points 2 2 1 2 4 5 2 4 4 7 1 4 2 5 3 1 Page 8