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Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000 Quiz 2 – Friday, September 10, 1999 NAME 0 1 0 1 2 ¡2 B B Let A = @ 2 0 1 C A, B = @ ¡1 1 1 1) The matrix B is a £ Box 1 Ã ! 3 2 ³ ´ 4 1 ¡2 1 4 C ,D= 3 1 4 A, C = 5 2 1 0 1 matrix. 2) Is it possible to multiply AB? Circle one: Yes or No. If Yes is your answer, what is the size of AB? 3) Is it possible to multiply BA? Circle one: Yes or No. If Yes is your answer, what is the size of BA? 4) Is it possible to multiply CD? Circle one: Yes or No. If Yes is your answer, what is the size of CD? 5) Is it possible to multiply BC? Circle one: Yes or No. If Yes is your answer, what is the size of BC? 6) Multiply the matrices B and D (you need to decide on the correct order). 7) Write the system of equations in matrix form; DON’T SOLVE. 2x ¡ 4y + 2z = 3 x ¡ 4y + 2z = 1 3x + 6y + 2z = 4 x ¡ y + 2z = 2 8) Apply the Gaussian Elimination process to the system 2x + y + 2z = 5 : Clearly show 3x + z =4 the new systems (or augmented matrices) which the Elimination process produces and clearly show the back substitution procedure. That is, be neat and orderly. In 9-13, assume that A is a 3 £ 3 matrix and that you are solving the system of equations ¡ ! ! A¡ x = b: 9) In general, list the three possibilities for the number of solutions . 0 1 0 1 0 1 0 1 ¤ ¤ ¤ ¤ B 10) After applying the Gaussian Elimination process, the augmented matrix is @ 0 0 ¤ ¤ C A 0 0 0 0 (a * means a non-zero number). How many free variables are there? Describe the solution space. ¤ ¤ ¤ ¤ B 11) After applying the Gaussian Elimination process, the augmented matrix is @ 0 ¤ ¤ ¤ C A 0 0 0 3 (a * means a non-zero number). How many free variables are there? Describe the solution space. ¤ ¤ ¤ ¤ C 12) After applying the Gaussian Elimination process, the augmented matrix is B @ 0 ¤ ¤ ¤ A 0 0 ¤ 0 (a * means a non-zero number). How many free variables are there? Describe the solution space. ¤ ¤ ¤ ¤ C 13) After applying the Gaussian Elimination process, the augmented matrix is B @ 0 0 0 0 A 0 0 0 0 (a * means a non-zero number). How many free variables are there? Describe the solution space. ¡ ! ! 14) Suppose that the output from Maple’s linsolve command when solving A¡ x = b is ¡21 ¡ _t1 + 5_t2 _t1 13 ¡ 3_t2 _t2 How many variables are in the system of equations? How many columns does the matrix A have? Is the system of equations homogeneous or non–homogeneous? Describe the solution space. The solutions are in what Rn space? ¡ ! Is it possible for b to be the zero vector? Write down two di¤erent solutions to the system of equations.