Monthly Archives: August 2011

Review exercise 1.29. Find 2 by 2 matrices that will (a) reverse the direction of a vector (b) project a vector onto the axis (c) rotate a vector counter-clockwise through 90 degrees (d) reflect a vector about the line that … Continue reading →

Review exercise 1.28. Compute the following matrices: Answer: For the first matrix we have Based on this, we can guess at the answer for raising the matrix to the power of , and try to prove it by induction. Assume … Continue reading →

Review exercise 1.27. State whether the following are true or false. If true explain why, and if false provide a counterexample. (1) If a matrix can be factored as where and are lower triangular with unit diagonals and and are … Continue reading →

Review exercise 1.26. (a) Given a 3 by 3 matrix what vector would make the product have 1 times column 1 of A plus 2 times column 3? (b) Construct a matrix for which the sum of column 1 and … Continue reading →

Review exercise 1.25. Given the matrix where what multiple of row 2 was subtracted from row 3 in elimination? Explain why is invertible, symmetric, and tridiagonal. What are the pivots? Answer: From the above we see that has been factored … Continue reading →

Review exercise 1.24. The equation defines a plane in 3-space. Find equations that define the following: (a) a plane parallel to the first plane but going through the origin (b) a second plane that (like the original plane) contains the … Continue reading →

Review exercise 1.23. Evaluate the following matrix expressions for and and then find the general expression for the first two matrices for any . Answer: We have and So the general equation appears to be We can prove this by … Continue reading →

Review exercise 1.22. Answer the following questions: (a) If has an inverse, does also have an inverse? If so, what is it? (b) If is both invertible and symmetric, what is the transpose of ? (c) Illustrate (a) and (b) … Continue reading →

Review exercise 1.21. Given the 2 by 2 matrix describe the rows of and the columns of . Answer: When multiplying from the left by to produce the first row of will be 2 times the first row of and … Continue reading →

Review exercise 1.20. The set of by permutation matrices constitute a group. (a) How many 4 by 4 permutation matrices are there? How many by permutation matrices are there? (b) For the group of 3 by 3 permutation matrices, what … Continue reading →