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mathematics
calculus

Questions and Answers of Calculus

Find a singular 2 × 2 matrix satisfying A2 = A.
Two competing companies offer satellite television to a city with 100,000 households. Gold Satellite System has 25,000 subscribers and Galaxy Satellite Network has 30,000 subscribers. (The other
The transpose of a matrix, denoted AT, is formed by writing its rows as columns. Find the transpose of each matrix and verify that (AB)T = BTAT.
Find x such that the matrix is equal to its own inverse.
1. Find x such that the matrix is singular.2. Verify the following equation.
1. Two matrices are ________ when their corresponding entries are equal. 2. When performing matrix operations, real numbers are usually referred to as ________. 3. A matrix consisting entirely of
Evaluate the expression.1.2. 3. 4.
Use the matrix capabilities of a graphing utility to evaluate the expression.1.2. 3. 4.
Solve for X in the equation, where1. X = 2A + 2B 2. X = 3A ˆ’ 2B 3. 2X = 2A - B
If possible, find AB and state the dimension of the result.1.2. 3.
Use the matrix capabilities of a graphing utility to find AB, if possible.1.2. 3. 4.
If possible, find(a) AB(b) BA(c) A2.1.2.
Solve for x and y.1.2. 3. 4.
Evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer.1.2.
Use matrices to find (a) u + v (b) u v (c) 3v u. 1. u = (1, 5), v = (3, 2) 2. u = (4, 2), v = (6, -3)
Find Av, where v = (4, 2), and describe the transformation.1.2. 3.
(a) Write the system of linear equations as a matrix equation, AX = B(b) Use Gauss-Jordan elimination on [A‹®B] to solve for the matrix X.1.2.
1. A corporation has four factories that manufacture sport utility vehicles and pickup trucks. The production levels are represented by A.Find the production levels when production increases by
1. A farmer grows apples and peaches. Each crop is shipped to three different outlets. The shipment levels are represented by A.The profits per unit are represented by the matrix B = [$3.50 $6.00].
1. A company has two factories that manufacture three sizes of boats. The numbers of hours of labor required to manufacture each size are represented by S.The wages of the workers are represented by
The matrixis called a stochastic matrix. Each entry pij (i ‰ j) represents the proportion of the voting population that changes from party i to party j, and pii represents the proportion
The numbers of calories burned by individuals of different body weights while performing different types of exercises for a one-hour time period are represented by A.(a) A 130-pound person and a
Determine whether the statement is true or false. Justify your answer. 1. Two matrices can be added only when they have the same dimension. 2. Matrix multiplication is commutative.
Use the matrices1. Show that (A + B)2 ‰ A2 + 2AB + B2. 2. Show that (A ˆ’ B)2 ‰ A2 ˆ’ 2AB + B2.
1. If a, b, and c are real numbers such that c ‰ 0 and ac = bc, then a = b. However, if A, B, and C are nonzero matrices such that AC = BC, then A is not necessarily equal to B. Illustrate
Find two matrices A and B such that AB = BA.
if possible, find(a) A + B,(b) A B(c) 3A(d) 3A 2B.1.2.
A corporation has three factories that manufacture acoustic guitars and electric guitars. The production levels are represented by A.(a) Interpret the value of a22. (b) How could you find the
1. Let A and B be unequal diagonal matrices of the same dimension. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products AB for several
1. If there exists an n × n matrix A−1 such that AA−1 = In = A−1A, then A−1 is the ________ of A. 2. A matrix that has an inverse is invertible or ________. A matrix that does not have an
Find the inverse of the matrix, if possible.1.2. 3.
Use the matrix capabilities of a graphing utility to find the inverse of the matrix, if possible.1.2. 3. 4
Find the inverse of the 2 Ã— 2 matrix, if possible.1.2. 3.
Use the inverse matrix found in Exercise 15 to solve the system of linear equations.1.2. 3. 4.
Use the inverse matrix found in Exercise 19 to solve the system of linear equations.1.2.
Use the inverse matrix found in Exercise 32 to solve the system of linear equations.1.2.
Use an inverse matrix to solve the system of linear equations, if possible.1.2. 3. 4.
Show that B is the inverse of A.1.2. 3. 4.
Use the matrix capabilities of a graphing utility to solve the system of linear equations, if possible.1.2.
You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 4.5% on AAA bonds, 5% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let x,
Consider the circuit shown in the figure. The currents I1, I2, and I3 (in amperes) are the solution of the systemwhere E1 and E2 are voltages. Use the inverse of the coefficient matrix of this system
Find the numbers of bags of potting soil that a company can produce for seedlings, general potting, and hardwood plants with the given amounts of raw materials. The raw materials used in one bag of
A florist is creating 10 centerpieces. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. The customer has a budget of $300 allocated for the centerpieces and wants each centerpiece
The table shows the numbers of visitors y (in thousands) to the United States from China from 2012 through 2014.(a) The data can be modeled by the quadratic function y = at2 + bt + c. Write a system
Determine whether the statement is true or false. Justify your answer. 1. Multiplication of an invertible matrix and its inverse is commutative. 2. When the product of two square matrices is the
Find the value of k that makes the matrix singular.1.2.
Consider matrices of the form(a) Write a 2 Ã— 2 matrix and a 3 Ã— 3 matrix in the form of A. Find the inverse of each. (b) Use the result of part (a) to make a conjecture about the
Consider the matrixUse the determinant of A to state the conditions for which (a) Aˆ’1 exists and (b) Aˆ’1 = A.
Verify that the inverse of an invertible 2 Ã— 2 matrix
Explain why the determinant of each matrix is equal to zero.a.b. 2. A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero. Find the determinant of each diagonal
Use the matrix capabilities of a graphing utility to find the determinant of the matrix.1.2. 3. 4. 5. 6.
Find all the(a) Minors(b) Cofactors of the matrix.1.2.
Find the determinant of the matrix. Expand by cofactors using the indicated row or column.1.(a) Row 1 (b) Column 1 2. (a) Row 2 (b) Column 2 3. (a) Row 2 (b) Column 2
Find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest.1.2. 3. 4.
Find the determinant of the matrix.1. [4]2. [- 10]3.4. 5. 6.
Use the matrix capabilities of a graphing utility to find the determinant.1.2. 3. 4.
Find(a) | A |(b) | B |(c) AB(d) | AB|1.2. 3.
Create a matrix A with the given characteristics. (There are many correct answers.) 1. Dimension: 2 × 2, |A| = 3 2. Dimension: 2 × 2, |A| = −5 3. Dimension: 3 × 3, |A| = −1 4. Dimension: 3 ×
Find the determinant(s) to verify the equation.1.2. 3.
Solve for x.1.2. 3. 4.
Find the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.1. 2. 3. 4.
Determine whether the statement is true or false. Justify your answer. 1. If a square matrix has an entire row of zeros, then the determinant of the matrix is zero. 2. If the rows of a 2 × 2 matrix
Find square matrices A and B such that |A + B| ≠ |A| + |B|.
Consider square matrices in which the entries are consecutive integers. An example of such a matrix isa) Use the matrix capabilities of a graphing utility to find the determinants of four matrices of
1. Describe the error.2. Let A be a 3 Ã— 3 matrix such that |A| = 5. Is it possible to find |2A|? Explain.
Explain why each equation is an example of the given property of determinants (A and B are square matrices). Use a graphing utility to verify the results.1. If B is obtained from A by interchanging
Use a determinant to find the area of the triangle with the given vertices.1.2. 3. (0, 4), (ˆ’2 ˆ’3), (2, ˆ’3)
Find a value of y such that the triangle with the given vertices has an area of 4 square units. 1. (−5, 1), (0, 2), (−2, y) 2. (−4, 2), (−3, 5), (−1, y)
1. A large region of forest is infested with gypsy moths. The region is triangular, as shown in the figure. From vertex A, the distances to the other vertices are 25 miles south and 10 miles east
Use a determinant to determine whether the points are collinear.1. (2, −6), (0, −2), (3, −8)2. (3, −5), (6, 1), (4, 2)3. (2, -1/2), (-4, 4), (6, -3)
Find the value of y such that the points are collinear.1. (2, −5), (4, y), (5, −2)2. (−6, 2), (−5, y), (−3, 5)
Use a determinant to find an equation of the line passing through the points.1. (0, 0), (5, 3)2. (0, 0), (−2, 2)3. (−4, 3), (2, 1)4. (10, 7), (−2, −7)
Use matrices to find the vertices of the image of the square with the given vertices after the given transformation. Then sketch the square and its image. 1. (0, 0), (0, 3), (3, 0), (3, 3);
Use a determinant to find the area of the parallelogram with the given vertices. 1. (0, 0), (1, 0), (2, 2), (3, 2) 2. (0, 0), (3, 0), (4, 1), (7, 1) 3. (0, 0), (−2, 0), (3, 5), (1, 5)
(a) Write the un coded 1 2 row matrices for the message, and then(b) Encode the message using the encoding matrix.Message............................ Encoding Matrix [1]
(a) Write the un coded 1 3 row matrices for the message, and then(b) Encode the message using the encoding matrix.Message…………………………. Encoding Matrix
Write a cryptogram for the message using the matrix
Use A-1 to describe the cryptogram.1.11 21 64 112 25 50 29 53 23 46 40 75 55 92 2. 85 120 6 8 10 15 84 117 42 56 90 125 60 80 30 45 19 26 3. 9 ˆ’1 ˆ’9 38 ˆ’19
Decode the cryptogram by using the inverse of A in Exercises 49-52. 1. 20 17 −15 −12 −56 −104 1 −25 −65 62 143 181 2. 13 −9 −59 61 112 106 −17 −73 −131 11 24 29 65 144 172
The cryptogram below was encoded with a 2 × 2 matrix. 8 21 −15 −10 −13 − 13 5 10 5 25 5 19 −1 6 20 40 −18 − 18 1 16 The last word of the message is _RON. What is the message?
The cryptogram below was encoded with a 2 × 2 matrix. 5 2 25 11 −2 −7 −15 − 15 32 14 −8 −13 38 19 −19 − 19 37 16 The last word of the message is _SUE. What is the message?
1. Consider the circuit shown in the figure. The currents I1, I2, and I3 (in amperes) are the solution of the systemUse Cramer's Rule to find the three currents.
A system of pulleys is loaded with 192-pound and 64-pound weights (see figure). The tensions t1 and t2 in the ropes and the acceleration a of the 64-pound weight are found by solving the system of
Determine whether the statement is true or false. Justify your answer. 1. In Cramer's Rule, the numerator is the determinant of the coefficient matrix. 2. Cramer's Rule cannot be used to solve a
1. Describe the error. Consider the systemThe determinant of the coefficient matrix is 2. At this point in the text, you know several methods for finding an equation of a line that passes through two
Use a determinant to find the area of the triangle whose vertices are (3, −1), (7, −1), and (7, 5). Confirm your answer by plotting the points in a coordinate plane and using the formula
Use Cramer's Rule (if possible) to solve the system of equations.1.2.3.
Determine the dimension of the matrix.1. [-13]2.3.4. [5]
Find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest.1.2.3.
Use Cramer's Rule (if possible) to solve the system of equations.1.2.
Write the system of linear equations represented by the augmented matrix. Then use back substitution to solve the system. (Use variables x, y, and z, if applicable.)1.2.
Use a determinant to find the area of the triangle with the given vertices.1.2.
Use a determinant to determine whether the points are collinear. 1. (−1, 7), (3, −9), (−3, 15) 2. (0, −5), (−2, −6), (8, −1)
Use a determinant to find an equation of the line passing through the points. 1. (−4, 0), (4, 4) 2. (2, 5), (6, −1)
Use a determinant to find the area of the parallelogram with the given vertices. 1. (0, 0), (2, 0), (1, 4), (3, 4) 2. (0, 0), (−3, 0), (1, 3), (−2, 3)
Decode the cryptogram using the inverse of the matrix1. ˆ’5 11 ˆ’2 370 ˆ’265 225 ˆ’57 48 ˆ’33 32 ˆ’15 20 245
Determine whether the statement is true or false. Justify your answer.1. It is possible to find the determinant of a 4 Ã— 5 matrix.2.
What is the cofactor of an entry of a matrix? How are cofactors used to find the determinant of the matrix?
Three people are solving a system of equations using an augmented matrix. Each person writes the matrix in row-echelon form. Their reduced matrices are shown below.Can all three be right? Explain.
Use matrices to solve the system of linear equations, if possible. Use Gaussian elimination with back-substitution.1.2.

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