MATRIX REVIEW PROBLEMS: Our matrix test will be on Friday May 23rd. Here are some problems to help you review.

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1 MATRIX REVIEW PROBLEMS: Our matrix test will be on Friday May 23rd. Here are some problems to help you review. 1. The intersection of two non-parallel planes is a line. Find the equation of the line. Give the equation in parametric form and in vector form. Hint: Use RREF, then let. a. 1st pair of planes b. 2nd pair of planes 2. Another way to write the equation of a line is to solve each of the parametric equations for t. This gives three expressions, all equal to t, and therefore all equal to each other. These are sometimes called symmetric equations of the line. Express the lines you found in 1a and 1b as symmetric equations. What do you notice about the denominators of the three fractions? 3. In the next three problems, you are given the equations of three planes. Determine the intersection of all three planes (this could be a point, a line, a plane or no intersection at all). If the intersection of all three is a line, find an equation for the line. a. 1st set of three planes b. 2nd set of three planes c. 3rd set of three planes 4. Derive the 2 by 2 matrix that rotates counter clockwise about the origin by. Remember: Find the images of the points and. These will be the columns of your matrix. 5. Suppose that A and B are 2-by-2 matrices where and. a. If, find. Hint: The determinant represents the area stretch factor. If you apply one transformation and then another, you are stretching the area and then stretching it again. b. Find. Explain how you got your answer. Hint: The inverse undoes the original transformation. How is this related to the idea from part a? 6. Suppose A and B are invertible. What is the determinant of? (Hint: How are the determinants of A and related?) Does this mean that? 7. a. Find the partial fraction decomposition of. In other words, find A and B such that b. Find the partial fraction decomposition of.. Note: If you re stuck, see section 7.4 in Demana.

2 8. Each of the following matrices represents a system of equations. Problems a c involve 3 variables and problem d has 4. Determine the solution set of each system. If the solution set is a line, find the equation of the line. If the solution set is a plane, find the equation of the plane. a. b. c. d. 9. Consider the following system of equations. a. Suppose the system of equations has a unique solution for x and y. What can you conclude about the value of k? b. In the above matrix, let k = 0. Then find the input point (x, y) whose image is (4, 2). c. Still assuming k = 0, consider vectors v = < 2, 2> and w = <1, 3>. Find the linear combination of v and w that equals <4, 2>. d. Still assuming k = 0, consider again, the vectors v = <-2, 2> and w = <1, 3>. Find the linear combination of v and w that equals <a, b>. [Your answer should be expressed in terms of a and b.] e. Using a method involving the determinant, find the area of the parallelogram with sides determined by v = < 2, 2> and w = <1, 3>. See page 462 of the Brown book for help if you are stuck.

3 10. Consider the linear transformation. The effect of this transformation can be reached using two of the basic transformations (reflection, rotation, dilation, translation). Identify which two. Be as specific as possible. Give the matrix for each transformation. Does the order of these transformations matter? 11. Suppose T is a transformation matrix that has the following effect. It first rotates the pre-image point clockwise, then reflects over the y-axis, and then compresses vertically by a factor of 2. Find T. 12. Give a geometric description of the following transformations. Be as specific as possible. (For example, if the transformation is a reflection, tell what plane the points are being reflected over). If you aren't sure what a transformation does, try a few points and see if you can identify a pattern. a. b. c. 13. Find matrices for each of the following 3D transformations. a. Reflect over the xy-plane. b. Stretch the x values by 3, stretch the y values by 5, and stretch the z values by 2. c. Rotate the yz-plane by direction of your choice. d. Rotate the xz-plane by direction of your choice. e. Rotate the xy-plane by counterclockwise. (Assume you are looking toward the origin from the +z direction which is pointing up from your paper.) 14. Solve the matrix equation for X. Your answer will be in terms of A, B, C, and D. 15. Find the matrix of a two dimensional linear transformation that reflects over the line.

4 ANSWERS: 1. a. Parametric: ; Vector: b. Parametric: ; Vector: 2. a. b. c. The denominators of the fractions are the direction vector of the line (i.e. the components of a vector parallel to the line). 3. a. No solution b. Line: c. Point: a. b. 6., but 7. a. b. 8. a. Line, b. No solution c. Point, (2, 3, 0) d. Line (in 4-space). All points of the form (4, 5, 7, w) 9. a. k cannot be 1 or 4. If k were equal to either 1 or 4, then the matrix would have a determinant of 0, resulting in either no solutions or infinite solutions. b. Since k is not one of the values you found in part a, the inverse of the matrix can be found:, so the input point is ( 3, 2). c. The multiplication in part b gives the answer: 3v 2w = <4, 2>. d., so ( )v + ( )w = <a, b>. e. Area = 4.

5 10. Reflect over the line, then stretch by 3 in the x direction and by 4 in the y direction. The matrices are followed by and The reflection is done first. The dilations can be done in either order. If you have another answer, make sure the matrices multiply to when multiplied in the correct order Remember to multiply in the correct order when finding T. 12. a. Reflection over the xz-plane b. Translation 3 units in positive x direction, 5 units in negative y direction and, 1 unit in negative z direction. (Not a linear transformation.) c. 90 rotation in the xz-plane 13. a. b. c. d. e