Algebra MAT1102 & Calculus I
Tutorial & Computer Laboratory: Week 11
Calculus: Finish Tutorial 10, and make sure you have completed all the Study Book activities for the sections covered in lectures. Algebra: This tutorial completes Chapter 2 on Matrices. By the end of this week you should have mastered the Study Book Readings for this chapter. 1. Write full solutions to the Homework problems listed below. Offer explanatory statements, use language and symbols carefully, and conclude clearly. Larson 2.1: Q 7,15,23,25. And 41,50,51 (Ed 4) or 43,54,55 (Ed 5). Larson 2.2: Q 9,17,18. And 27,28,29,31,32,39 (Ed 4) or 29,30,31,33,34,41 (Ed 5). Larson 2.3: Q 3,13,25, and 35, 37 (Ed 5) (33,35, Ed 4) Larson 2.5: One exercise of your choice. 2. Find matrices A and B for which AB = BA. Offer one pair of different sizes, one pair of the same size.
3. Write a 2 4 matrix M of your choice. What size identity matrices I will make M I = M and IM = M ?. Check by hand.
4. Matrix algebra follows many of the rules we expect, and we can use those to rearrange and solve equations. However, AB = BA in general, and we need to take care not to be careless! (a) Why does the matrix expression: (2A - B)(3B - A) = not yield middle term 7AB? (b) Why is (AB)n not the same as An B n , in general?
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5. (a) Find the 3 3 matrix A, given 3A - 5I3 = 9I3 + A, I3 the identity matrix. (b) Given that matrix A is non-singular (ie A-1 exists) and A, B and X are sized appropriately for each equation below, solve each for X. Take care to left- or right-multiply by A-1 so that it is adjacent to A. AX = B XA = B , AX + C = B.
(a) 3A - 5I3 = 9I3 + A gives
(b) AX = B gives XA = B gives AX + C = B gives
6. (a) Given C =
1 2
1 4
, reduce [C : I] by hand to find the inverse of C.
(b) Check that you get the same matrix using the fact that the inverse of and confirm by hand that CC
-1
a c
b d
is
1 ad-bc
d -b -c a
,
=I=C
-1
C.
(c) Write a 2 2 matrix D that does not have an inverse. Use the denominator ad - bc (called the determinant) to guide your choice.
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7. Write the systems below in the form Ax = b. Briefly describe the contents of the matrices A, x and b. a + 2b - c = 1 b + 3c = 3 2c = 4
x1 - 2x2 - x3 - 2x4 = 0 3x1 - 5x2 - 2x3 - 3x4 = 1 2x1 - 5x2 - 2x3 - 5x4 = -1 - x1 + 4x2 + 4x3 + 11x4 = 2
Prepare to solve these equations by two methods when you are in the Lab: Firstly write the two augmented matrices that you will row-reduce. Now write the single matrix equation you can use to solve for X if A-1 exists. Show carefully that you need to pre-multiply and not post-multiply by A-1 to be able to use A-1 A = I to get X = A-1 B. What Matlab commands will this do calculation for you?
8. If the linear system AX = B has A 3 5 and B a column matrix, how many variables are there? Give the dimensions of X, B, and the augmented matrix for the system. Explain clearly why you cannot use the inverse of A to solve this system.
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9. Consider the matrix equation M X = N where M is invertible, and N is a given 4 7 matrix. Say how you know that there will be a unique solution X, give the form of the solution, and the size of X.
In the Lab: IMPORTANT: Using inverses to solve matrix equations: You know how to solve the systems of equations Ax = b by row-reducing the matrix [A the coefficient matrix and b is the column of RHS's. You also know that if A-1 exists, you can solve the system Ax = b using x = A-1 b. Solve the two linear systems in Q 7 by both methods. First row-reduce the augmented matrix, record the output, and write the solutions using the correct variables. Then solve for X using A-1 b. Note that you should enter the coefficient matrix A and the column b first, or enter b as a row, and use the command b' to transpose it into a column. This means you can enter the coefficient matrix A, and b as a row, row-reduce the augmented matrix with the command rref([A b']) . Then check your answers using x =inv(A)*b' Make sure you understand both methods and these commands. b] where A is
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Note that MATLAB offers a third method to solve a linear system, Ax = b: The command A\b employs a least squares type of process which we will study in Alg & Calc II. It is best read as "A divides b" or "A divides into b". Not...