# LA23sampletest2

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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