Monthly Archives: July 2011

Review exercise 1.19. Solve the following systems of equations using elimination and back substitution:    and    Answer: We start with the system and subtract 1 times the first equation from the second and 3 times the first equation from … Continue reading →

Review exercise 1.18. Given the following matrix (a) Factor into the form (if ). (b) Show that exists and has the same form as . Answer: (a) We start elimination by subtracting times the second row from the third row … Continue reading →

Review exercise 1.17. Factor the following two symmetric matrices into the form . Answer: We start elimination for the first matrix by subtracting 2 times the first row from the second row (): and then subtract 2 times the second … Continue reading →

Review exercise 1.16. Given the following system of equations: what must be for the system to have no solution? One solution? An infinite number of solutions? Answer: If then the system reduces to for which and is (obviously) a solution. … Continue reading →

Review exercise 1.15. For the following by matrix and its inverse what is the value of ? Answer: Since we must have Multiplying both sides by we obtain or so that . NOTE: This continues a series of posts containing … Continue reading →

Review exercise 1.14. For each of the following find a 3 by 3 matrix such that for any matrix (a) (b) (c) The first row of is the last row of and the last row of is the first row … Continue reading →

Review exercise 1.13. Given the following: use the triangular systems and to find a solution to Answer: We have Since and we have also. Since we have so that We then have Since and we have . We then have … Continue reading →

Review exercise 1.12. State whether the following are true or false. If a statement is true explain why it is true. If a statement is false provide a counter-example. (a) If is invertible and has the same rows as but … Continue reading →

Review exercise 1.11. Suppose is a 2 by 2 matrix that adds the first equation of a linear system to the second equation. What is ? ? ? Answer: Since adds the first equation to the second, we have We … Continue reading →

Review exercise 1.10. Find the inverse of each of the following matrices, or show that the matrix is not invertible. Answer: We use Gauss-Jordan elimination on the first matrix , starting by multiplying the first row by 1 and subtracting … Continue reading →