Exam 3-Review-filled in PDF

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Exam 3 - Review Chapter 5: Systems of linear equations • lines, slope, intercepts(to find the x-intercept, set y = 0 and solve x, to f ind the y-intercept, set x = 0 and solve y) point-slope form (y − y1 = m(x − x1 )), slope-intercept form (y = mx + b), parallel lines (same slope), • solving linear systems using substitution • systems of linear equations have no solution (inconsistent), an unique solution, or infinitely many solutions (consistent) • A matrix is in the REDUCED FORM if : – The entry in the first row and first column of the coefficient matrix is a 1, and the first nonzero entry in each row is a 1. These entries are called leading ones (1s). – If a column of the coefficient matrix contains a leading 1, then all other entries in that column are 0. – The leading 1s march downward and to the right (as you move from left to right through the columns of the coefficient matrix, the leading 1s occur in succesive rows). All rows containing only zeros are at the bottom. • solving linear systems using reduction method: – write augmented matrix – reduction operations: (1) interchange of rows, (2) multiplication of a row by a nonzero number, (3) subtracting or adding a multiple of one row from another row

2 – use the operations above to bring the augmented matrix into reduced form – interpret the results. • How to determine the number of solutions of a system of equations: – Case 1: If the reduced form has a row with zeros on the left of the vertical line and a nonzero entry on the right, then the system is inconsistent, i.e. it has no solution. – Case 2: If a system is consistent (no inconsisten rows) and the reduced form has the same number of variables as the number of nonzero rows, then the system has a unique solution. – Case 3: If a system is consistent and the number of variables is larger than the number of nonzero rows of the reduced form matrix, then the system has infinitely many solutions (in other words the reduced form has columns without a leading 1). The variables associated with the columns not containing leading 1s can be specified arbitrarily (those variables are called arbitrary parameters).The number of arbitrary parameters equals the number of columns that do not contain a leading 1.

3 Practice problems: For each of the augmented matrices in the next three problems, determine which of the following statements is true about the associated system of linear equations: (A)The system has no solution. (B) The system has exactly one solution (C) The system has exactly four solutions. (D) The system has infinitely many solutions with ONE arbitrary parameter. (E) The system has infinitely many solutions with TWO arbitrary parameters. (N) none of the above. 1.

3 2 1 0 4 −2 5 40 1 2 −4 35 0 2 4 −8 5

2. 2

3 1 0 1 3 2 6 0 1 −3 4 1 7 6 7 42 0 2 6 4 5 0 −1 3 −4 −1

4 3. 2 1 60 6 41 0

0 1 1 −1 0 2 1 −1

0 3 0 4

3 4 27 7 75 5

5 4. Write the complete solution set to the linear system represented by the augmented matrix 2

3 2 2 4 1 2 −2 4 0 0 2 −1 2 4 5 0 0 0 1 0 2

Let the variables for the problem be v, w, x, y, and z.

6 5. Find the equation for the straight line that goes through (6,4) and is parallel to the line 6x − 3y = 1.

7 6. Arnold’s backpack holds 270 cubic inches of supplies, and he has $146 to spend on supplies at the local otfitter. He only needs two things: dehydrated camp dinners, each of which costs $8 and takes up 10 cubic inches of space; and water bottles, each of which costs $5 and takes up 15 cubic inches of space. How many camp dinners and water bottles should Arnold buy if he wants to spend all of his money and use all of the available space in his backpack.

8 7. The Corner Deli uses a linear relationship, based on length, to determine the selling price of a sandwich. If a 5 inch sandwich sells for $3.50 and an 8 inch sandwich sells for $5.30, what is the price of a 2-foot sandwich? (1foot=12 inches).

9 Chapter 6: Matrices • Operations with matrices: addition, subtraction, scalar multiplication, row-column product, matrix multiplication • Identity matrix I: square matrix, with 1’s on the diagonal, and 0’s elsewhere • Matrix multiplication is not commutative! • Inverting 2×2 matrix by formula :     d −b 1 a b −1 – The inverse of A = . is the matrix A = c d ad − bc −c a – If ad − bc = 0, then the matrix has no inverse. • inverting matrices using row reduction : – Form the augmented matrix [A|I]. – Row reduce this matrix until A is brought into reduced form R, so that the augmented matrix has the form [R|B]. – If R = I, then A is invertible and has inverse B, otherwise A is not invertible. • the Leontief Input-Output model: A is a n × n matrix (technology matrix), X (production schedule) and D (demand vector) are column n-vectors (keep in mind that the entry (i,j) of A is the amount of Gi required to produce one unit of Gj . -production schedule: X = (I − A)−1 D or row reduce [I − A|D] -demand vector: D = (I − A)X

10 Practice problems: 1. Find the external demand satisfied by the technology matrix and production schedule given below: 3 3 2 2 .5 .8 .1 220 A = 4 0 .2 .45 X = 4 1005 . .1 0 .4 120

11 2. An economy produces two goods: silver and gold. The production of one ounce of silver requires .1 ounce of silver and .2 ounce of gold, while the production of one ounce of gold requires .3 ounce of silver and .4 ounce of gold. There is an external demand for 5 ounces of silver and 6 ounces of gold. Find the production schedule for this economy.

12 3. If Q is a 3×6 matrix, R is a 4×4 matrix, and T is a matrix for which QTR is defined, then what dimension must T have?

4. Find the inverse of

2 3 1 2 −3 A = 40 1 −2 5 1 1 0

13 5. Find the entry in the second row and second column of AB − 2C, where       5 −1 2 3 −1 1 −1 2 , C= , B= A= −3 2 0 0 −2 −2 3 −4

6. Let

  1 −1 A= , 2 2

  1 1 , B= 2 c

Find the value of c such that AB does not have an inverse.

14 Chapter 8: Markov chains

• states, transitions, Markov property: probability of moving from a state to any state on the next observation depends only on the two states, and not on what happened earlier. • transition matrix, transition diagram • multi-step transition matrix, P(k) = Pk ; multi-step transition probabilities, pij (k) • probability vector (sum of the entries is 1); state vector after k transitions : Xk = X0 Pk , X0 is called initial state vector. • regular (some power of the transition matrix has only positive entries, no 0’s - it is enough to check the first (N − 1)2 + 1 powers) and non-regular Markov chains • finding stable vector and stable probabilities of a regular Markov chain by solving the system: W(P − I) = O N X

wi = 1

i=1

Tips: - when some of the entries in (P − I) are 0, substitution method works well; when all the entries are nonzero substitution method is harder to use, you may want to use row reduction. -Transform all decimals in fractions.

15 Practice Problems: 1. A Markov chain has a transition matrix   1/4 3/4 P= 1/2 1/2 If the system is twice as likely to be in state 1 as to be in in state 2 on the first observation, what is the probability that the system is in state 2 on the third observation?

16 2. Kris has a quiz in her M118 class each week. If she gets a good score one week, then the probability she gets a good score the next week is 4 times the probability that she gets a poor score, and if she gets a poor score one week then the probability she gets a poor score next week is 2 times higher than the probability she gets a good score. Assume that her performance on quizzes, good or poor, can be described by a Markov chain. Let State 1 be a good score and State 2 be a poor score. Find the transition matrix.

17 3. Which of the following Markov chains are regular: 2 3 0 .6 .4 P1 = 4 1 0 0 5 0 1 0

3 2 .4 .2 .4 P2 = 4 0 1 0 5 0 .1 .9

3 2 0 .7 .3 P3 = 41 0 0 5 1 0 0

18 4. Find the third coordinate in the vector of stable probabilities for the Markov chain with transition matrix: 2

3 0 1 0 P = 4 0 0 15 .2 0 .8

Note: The above problem can also be formulated as: ”Find the probability that the system will be in state 3 in the long run.”...