Review - Ch. 1 & 2: Systems of Linear Equations & Matrices

The equation is not linear in the variables x and y.

LARLINALG8 1.1.005. Determine whether the equation is linear in the variables x and y. 3 sin x − y = 16

x = 1 - s - t, y = s, z = t

LARLINALG8 1.1.009. Find a parametric representation of the solution set of the linear equation. (Enter your answer as a comma-separated list of equations. Use s and t as your parameters.) x + y + z = 1

-5, 0

LARLINALG8 1.1.019. Graph the system of linear equations. Solve the system. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set y = t and solve for x in terms of t.)

(7/2) - (t/2), 6t - 1, t

LARLINALG8 1.1.051. SHOW YOUR WORK Solve the system of linear equations. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, and x3 in terms of the parameter t.)

NO SOLUTION

LARLINALG8 1.1.053. SHOW YOUR WORK Solve the system of linear equations. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set z = t and solve for x and y in terms of t.)

There are no rows consisting entirely of zeros, No, There exists at least one row which does not have a leading 1, No, neither

LARLINALG8 1.2.023.SBS. Determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. STEP 1: Check rows consisting entirely of zeros. Do all rows (if any) consisting entirely of zeros occur at the bottom of the matrix? __________ STEP 2: Check the first nonzero entry of each row. Does each row that does not consist entirely of zeros have the first nonzero entry equal to 1? __________ STEP 3: Check successive nonzero rows.If each nonzero row has a leading 1, is the leading 1 in the higher row farther to the left of the leading 1 in the lower row for each pair of successive rows? __________ STEP 4: Check the columns with leading ones. Does every column with a leading 1 have zeros in every position above and below its leading 1? __________ STEP 5: Determine the form of the matrix. __________

(2, -5)

LARLINALG8 1.2.029. Solve the system. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set y = t and solve for x in terms of t.)

5, -4, 3

LARLINALG8 1.2.031. SHOW YOUR WORK Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, set x3 = t and solve for x1 and x2.)

-t, s, 0, t

LARLINALG8 1.2.045. Solve the homogeneous linear system corresponding to the given coefficient matrix. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, x3, and x4 in terms of the parameters t and s.)

x1 + x2 + x3 = 0, x1 + x2 + x3 = 1

LARLINALG8 1.2.061. Is it possible for a system of linear equations with fewer equations than variables to have no solution? If so, give an example.

-4, ½

LARLINALG8 1.2.066. Find all values of λ (the Greek letter lambda) for which the homogeneous linear system has nontrivial solutions. (Enter your answers as a comma-separated list.) (2λ + 7)x − 4y = 0 x − λy = 0

(700 - s - t, 300 - s - t, 100 - t, t); (660, 260, 0, 60, 40); (580, 180, 60, 40, 60)

LARLINALG8 1.3.021. SHOW YOUR WORK The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for xi, i = 1, 2, , 5. (If the system has an infinite number of solutions, express x1, x2, x3, x4, and x5 in terms of the parameters s and t.) (b) Find the traffic flow when x3 = 0 and x5 = 40. (c) Find the traffic flow when x3 = x5 = 60.

[0, 4, 3, -2]; [-8, -1, 36, 6]

LARLINALG8 2.1.015. Find, if possible, AB and BA. (If not possible, enter IMPOSSIBLE.)

-3, -2, -1

LARLINALG8 2.1.045. Write the system of linear equations in the form Ax = b and solve this matrix equation for x. (Enter your answer for x as a comma-separated list)

[2, 3, 11, -1, 0, 2]

LARLINALG8 2.2.019. Perform the indicated operations, given: (B + C)A (Enter your answer as a comma-separated list, and surround with brackets.)

[13/200, 1/20, 2/25, 13/200]; [13/200, 2/25, 1/20, 13/200]

LARLINALG8 2.2.026. Show that AB and BA are not equal for the given matrices. AB: __________ BA: __________

[-5, 3, 2, 2, 6, 1, -6, 6, 3]; [-5, 3, 2, 2, 6, 1, -6, 6, 3]; Yes

LARLINALG8 2.2.043.SBS. Verify that (AB)^T = (B^T )(A^T). STEP 1: Find (AB)^T. __________ STEP 2: Find (B^T )(A^T). __________ STEP 3: Are the results from Step 1 and Step 2 equivalent? __________ (Separate matrix entries with a comma, STEPS with a semicolon, and surround each matrix with brackets)

[10, -3/5, 0, 20, -6/5, 10, 0, -1/5, 10]

LARLINALG8 2.3.022. SHOW YOUR WORK Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.)

-63/2

LARLINALG8 2.3.056. Find x such that the matrix is singular.

[1/2, 1/2, -1/2, -1/4]

LARLINALG8 2.3.057. Find A.

The matrix is elementary. It can be obtained from the identity matrix by interchanging two rows.

LARLINALG8 2.4.005. Determine whether the matrix is elementary. If it is, state the elementary row operation used to produce it.

[0, 0, 1, 0, 1, 0, 1, 0, 0]

LARLINALG8 2.4.009. Find an elementary matrix E such that EA = B.

[1, 0, 1, 1]; [1, -1, 0, 1]; [1, 0, 0, -3]

LARLINALG8 2.4.029. Find a sequence of elementary matrices whose product is the given nonsingular matrix.

[1, 0, 0, 0, 1, 0, -a/c, -b/c, 1/c]

LARLINALG8 2.4.039. SHOW YOUR WORK Use elementary matrices to find the inverse of A.

[1, 0, 0, 4, 1, 0, -1, 1, 1]; [3, 0, 1, 0, 1, -3, 0, 0, 4]

LARLINALG8 2.4.045. Find the LU-factorization of the matrix. (Your L matrix must be unit diagonal.)

b = 0, a = 0; b = 0, a = any real number

LARLINALG8 2.4.053. Determine a and b such that A is idempotent. (Select all that apply.)

MEET_ME_TONIGHT_RON_

LARLINALG8 2.6.007. The cryptogram below was encoded with a 2 × 2 matrix. 44 31 35 30 13 13 15 10 75 55 51 37 29 22 60 40 69 51 42 28 The last word of the message is RON_. What is the message?

13 -26 21 33 -53 -12 18 -23 -42 5 -20 56 -24 23 77

Use the invertible matrix to encode the message MEET ME MONDAY.