LA23sampletest2

Math 2050 Sample problems for midterm test 2 July 2023 Instructor Margo Kondratieva PLEASE NOTE The midterm is scheduled on July 14 at 2pm in ED1020 . Duration 50 minutes. 1. Use a dark pencil or pen and write clearly to make the work legible. 2. Show all your work. Write your answers in the indicated spots. 3. Any calculators and books/notes are NOT permitted at the exam. 1. Express ⃗v = " 0 2 # as a linear combination of the columns of A = " 1 1 3 2 # Answer : 2. Is the following set of vectors linearly dependent or independent? Explain.    10 20 30    ,    10 20 10    ,    30 60 50    , Answer : 3. True or False? (a) For any pair of 3X3-matrices A and B , AB ̸ = BA . (b) For all 3X3-matrices A and B , ( A + B ) T = B T + A T = A T + B T . (c) Let A and B be two N × N -matrices, N ≥ 2, such that AB=0. Then either A = 0 or B = 0. Here 0 represents a zero N × N -matrix. (d) For any four 3X3-matrices, ( ABC ) D = A ( BCD ). 4. Consider matrices A = " 1 − 1 3 2 # , B = " 2 1 1 2 # . a) Find AB , A T , A + B , and 2 AB . b) Find X given that XA = X + B .

5. Consider the following system of linear equations      x + y + z = 0 2 x − y − z = − 3 2 y + 4 z = 4 (a) rewrite the system in the matrix form AX=B Answer : A = , X = , B = (b) solve the system Answer : (c) Complete the sentence by circling what is appropriate: "This system of equations geometrically represents..." - three lines intersecting at a point - three planes intersecting at a line - three planes intersecting at a point - two planes intersecting with a line - none of the above (d) express vector    0 − 3 4    as a linear combination of the columns of matrix A from (a) 6. Consider the following augmented matrix    1 0 − 3 4 5 1 0 1 2 0 1 − 3 0 0 0 2 0 4    Solve corresponding system of linear equations and write the solution in the vector form. Answer : 2

7. The augmented matrix of a linear system is carried to    1 − 2 3 − 3 0 1 k + 25 2 0 0 9 − k 2 k + 3    by elementary row operations. For what value(s) of k does the system have (a) infinitely many solutions? (b) no solutions? (c) unique solution? 8. Consider A =    0 1 1 1 2 0 1 − 1 1    . Use Gaussian elimination method (elementary row operations) to find A − 1 , if it exists. 9. Find the inverse of the following elementary matrices:      1 0 0 0 0 2 0 0 0 0 1 0 0 0 0 1      ,      0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1      ,      1 0 0 5 0 1 0 0 0 0 1 0 0 0 0 1      10. Find LU-factorization for matrix A =    1 1 2 0 1 1 − 1 3 5    . 11. Solve the system ( LU ) X = B , where B =    0 − 3 4    , L =    1 0 0 2 1 0 − 1 3 1    and U =    1 1 2 0 1 − 2 0 0 1    3