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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Make the indicated changes in the determinant at the right, and then solve the indicated problem. Assume the elements are nonzero, unless otherwise specified.By what factor is the value of the
Find the BA−1. B = 8 -2 3 4 5 2 -3 -1 0 -2 1 04
Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix A, A2 = A × A.Show that A2 = A. A = 2-3-5 4 1-3-4 - 1 10 5 B = 1-2-6 -3 2 20 9 C 0-3 -|· 1
Find the CA−1. B 8-2 3 C 5 -1 0 2 -2 1 -3 04
Find the indicated matrices using a calculator.5A − 4B A = c = C 6-3 4 -5 B = -1 4 -7 2 -6 11 3 -9 D = 12 -6 79-6 -4 0 8
Solve any pair of the system of equations 2x − y = 4, 3x + y = 1, and x − 2y = 5. Show that the solution is valid for any pair chosen. What conclusions can be drawn about the graphs of the three
Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix A, A2 = A × A.Show that C2 = O. A = 2-3-5 4 1-3-4 - 1 10 5 B = 1-2-6 -3 2 20 9 C 0-3 -|· 1
Two jets are 2370 km apart and traveling toward each other, one at 720 km/h and the other at 860 km/h. How far does each travel before they pass?
Solve the given systems of equations using the inverse of the coefficient matrix.2x − 3y = −94x − y = −13
Make the indicated changes in the determinant at the right, and then solve the indicated problem. Assume the elements are nonzero, unless otherwise specified.How is the value changed if a is added to
Find the CA−1. B 8-2 3 C 5 -1 0 2 -2 1 -3 04
Find the indicated matrices using a calculator.−4C − 3D A = c = C 6-3 4 -5 B = -1 4 -7 2 -6 11 3 -9 D = 12 -6 79-6 -4 0 8
In solving the system of equations 3x − 4y = 5, 8y − 6x = 7, when conclusion can be drawn?
Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix A, A2 = A × A.Show that B3 = B. A = 2-3-5 4 1-3-4 - 1 10 5 B = 1-2-6 -3 2 20 9 C 0-3 -|· 1
Forces F1 and F2 hold up a beam that weights 2540 N, as shown in Fig. 16.14. The equations used to find the forces areA sin 47.2°+ B sin 64.4° = 2540A cos 47.2° − B cos 64.4° = 0Find the
The voltage across an electric resistor equals the current (in A) times the resistance (in Ω). If a current of 3.00 A passes through each of two resistors, the sum of the voltages is 10.5 V. If 2.00
Find the indicated matrices using a calculator.−6B − 4A A = c = C 6-3 4 -5 B = -1 4 -7 2 -6 11 3 -9 D = 12 -6 79-6 -4 0 8
In applying Kirchhoff’s laws to the circuit shown in Fig. 16.21, the following equations are found. Determine the indicated currents (in A). IA +IB +IC+ID+1E = 0 -21A +31B = 0 31B - 31 c = 6 -31c +
In applying Kirchhoff’s laws to the circuit shown in Fig. 16.15, the following equations are found. Determine the indicated currents (in A).Fig. 16.15. Ic IA + B + IC = 0 = 6 21 A 51 B 51B - Ic = -3
Solve the given systems of equations using the inverse of the coefficient matrix.5A − 7B = 626A + 5B = −6
Three machines together produce 650 parts each hour. Twice the production of the second machine is 10 parts/h more than the sum of the production of the other two machines. If the first operates for
Find the CA−1. B 8-2 3 C 5 -1 0 2 -2 1 -3 04
Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix A, A2 = A × A.Show that A3B3 = AB. A = 2-3-5 4 1-3-4 - 1 10 5 B = 1-2-6 -3 2 20 9 C 0-3 -|· 1
In analyzing the forces A, B, C, and D shown on the beam in Fig. 16.22, the following equations are used. Find these forces. A + B = 850 A + B + 400 = 0.8C + 0.6D 0.6C = = 0.8D 5A 5B + 4C - 3D =
Solve the given systems of equations using the inverse of the coefficient matrix.33x + 52y = −45045x − 62y = 1380
Find the indicated matrices using a calculator.3C − 2D A = c = C 6-3 4 -5 B = -1 4 -7 2 -6 11 3 -9 D = 12 -6 79-6 -4 0 8
A business bought three types of computer programs to be used in offices at different locations. The first costs $35 each and uses 190 MB of memory; the second costs $50 each and uses 225 MB of
Use matrices A and B to show that the indicated laws hold for these matrices, explain the meaning of the result.A + B = B + A A || -1 2 3 0 -3 -3 -1 9-10-2 7 4 B = 4 5 1 -1 -3 0 0-1 1 11 8 2
For matricesshow that AB = BA. A = a b ba and B = c d d c
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by(If the answer comes out negative, take the absolute value). A natural gas company locates the three corners of
Solve the given systems of equations using the inverse of the coefficient matrix.0.24x − 0.26y= −3.10.40x+ 0.34y = −1.3
Two batteries in an electric circuit have a combined voltage of 18 V, and one battery produces 6 V less than twice the other. What is the voltage of each?
Use matrices A and B to show that the indicated laws hold for these matrices.A + O = A A 7 4 -1 2 3 0-3 -1 9-10-2 B = = 4 5 1 -1 -3 0 0 -1 1 11 8 2
Find the determinant of the matrixExplain what this tells us about its inverse. 1-2 0 -2 48 3 -6 6
For matrixshow that A2 − 4A − 5I = O. A = T 212 22 2 2 1
Solve the given systems of equations using the inverse of the coefficient matrix.2u − 3v + 2w = 7−3u + v − 3w = −6u + 4v + w = −13
For college expenses, a student took out a loan at 4.00%, and a semester later took out a second loan at 3.00%. The total annual interest for the two loans was $245. If the second loan had been for
Use matrices A and B to show that the indicated laws hold for these matrices.−(A − B) = B − A A -1 2 3 0 -3 -1 9-10-2 7 4 B = 4-1 -3 0 0 -1 1 11 8 2 5 1
For the matrixShow that This verifies the method of Example 1.Data from Example 1Find the inverse of the matrixFirst, we interchange the elements on the principal diagonal and change the signs
An alloy is to be made from four other alloys containing copper (Cu), nickel (Ni), zinc (Zn), and iron (Fe). The first is 80% Cu and 20% Ni. The second is 60% Cu, 20% Ni, and 20% Zn. The third is 30%
Solve the given systems of equations using the inverse of the coefficient matrix.6x + 6y − 3z = 24x + 4y + 2z = 53x − 2y + z = 17
Forwhere j = √−1, show that J 2 = −I, J3 = −J, and J4 = I. Explain the similarity with j2 , j3, and j4. j o J oj
What volume of each of a 20% acid solution and a 50% acid solution should be combined to form 48 mL of a 25% solution?
Describe the relationship between the elements of the matrix and the elements of its inverse. a 0 0 090 0 0 с
Solve the given problem. Using two rows and columns, show that (−I)2 = I.
In testing for air pollution, a given air sample contained a total of 6.0 parts per million (ppm) of four pollutants, sulfur dioxide (SO2), nitric oxide (NO), nitrogen dioxide (NO2), and carbon
Use matrices A and B to show that the indicated laws hold for these matrices.3(A + B) = 3A + 3B A -1 2 3 0 -3 -1 9-10-2 7 4 B = 4-1 -3 0 0 -1 1 11 8 2 5 1
Solve the given systems of equations using the inverse of the coefficient matrix.x + 2y + 3z = 13x − 4y − 3z = 27x − 6y + 6z = 2
In Exercise 44 of Section 16.2, we saw that the matrix multiplicationrotates the points in the second matrix counterclockwise about the origin by angle θ. If we replace the first matrix with its
Two computer programs together require 236 MB (megabytes) of memory. If one program requires 20.0 MB of memory more than twice the memory of the other, what are the memory requirements of each
Show that A2 − I = (A + I)(A − I) for A = 24 35
A tablet with a 32-GB hard drive starts out with three different apps A, B, and C, which use up 4% of the tablet’s memory. Two more apps are added, each using the same amount of memory as app A,
The matrixis symmetric (note the elements on opposite sides of the main diagonal are equal). Show that A−1 is also symmetric. A 2 -1 1 -1 4-3 1-3 2
Find the unknown quantities in the given matrix equations. An airplane is flying in a direction 21.0° north of east at 235 km/h but is headed 14.5° north of east. The wind is from the southeast.
Solve the given systems of equations using the inverse of the coefficient matrix.3x + 2y + z = 22x + 3y − 6z = 3x + 3y + 3z = 1
By using Show that AB = AC does not necessarily mean that B = C. 3 A-1880-1280-128) B = 63
Solve the given systems of equations by Gaussian elimination.Exercise 31Data from Exercises 31Solve the given systems of equations using the inverse of the coefficient matrix.2x − 3y = −94x − y
Find the electric currents shown in Fig. 16.3 by solving the following matrix equation:Fig. 16.3. 1₁ + 12 + 13 -21, +31₂ -31₂ + 613 2 0 24 0
A research chemist wants to make 10.0 L of gasoline containing 2.0% of a new experimental additive. Gasoline without additive and two mixtures of gasoline with additive, one with 5.0% and the other
From past records, satellite television Company A finds that among its current customers, 92% will still be a customer one year from now and the remaining 8% will be lost to other competing
For the four-terminal network shown in Fig. 16.9, it can be shown that the voltage matrix V is related to the coefficient matrix A and the current matrix I by V = A−1I, whereFind the equations for
The contractor of a housing development constructs four different types of houses, with either a carport, a one-car garage, or a two car garage. The following matrix shows the number of houses of
Solve the given systems of equations by Gaussian elimination.Exercise 32Data from Exercises 32Solve the given systems of equations using the inverse of the coefficient matrix.5A − 7B = 626A + 5B =
A river tour boat takes 5.0 h to cruise downstream and 7.0 h for the return upstream. If the river flows at 4.0 mi/h, how fast does the boat travel in still water, and how far downstream does the
The inventory of a drug supply company shows that the following numbers of cases of bottles of vitamins C and B3 (niacin) are in stock: Vitamin C—25 cases of 100-mg bottles, 10 cases of 250-mg
Solve the given systems of equations by Gaussian elimination.Exercise 35Data from Exercises 35Solve the given systems of equations using the inverse of the coefficient matrix.2u − 3v + 2w = 7−3u
In Exercise 41, the market share of Company A in future years can be found by multiplying by the transition matrix T repeatedly. What would the market share of Company A be after 3 years?Data from
In studying the motion of electrons, one of the Pauli spin matrices used iswhere j = √−1. Show that sy2 = I. 0 -j j 0
The rotations of a robot arm such as that shown in Fig. 16.10 are often represented by matrices. The values represent trigonometric functions of the angles of rotation. For the following rotation
An avid math student sends an encrypted matrix message E to a friend who has the coding matrix C. The friend can decode E to get message M by multiplying E by C−1. Ifwhat is the message? (0 =
To rotate a set of points (x1, y1), (x2, y2), (x3, y3), . . . counterclockwise about the origin by angle θ, we multiply If a photo on a computer screen has corners at (4, 2), (−4, 2),
Solve the given systems of equations by Gaussian elimination.Exercise 36Data from Exercises 36Solve the given systems of equations using the inverse of the coefficient matrix. 6x + 6y − 3z =
In an ammeter, nearly all the electric current flows through a shunt, and the remaining known fraction of current is measured by the meter. See Fig. 16.5. From the given matrix equation, find voltage
One serving of brand K breakfast cereal provides the given percentages of the given vitamins and minerals: vitamin A, 15%; vitamin C, 25%; calcium, 10%; iron, 25%. One serving of brand G provides:
Solve the given systems of equations by Gaussian elimination.Exercise 37Data from Exercises 37Solve the given systems of equations using the inverse of the coefficient matrix. x + 2y + 3z = 13x
Using the same encryption code C, the student’s friend answered with What was the return message? (See Exercise 43.) Data from Exercises 43An avid math student sends an encrypted matrix
In the theory related to the reproduction of color photography, the equationis found. The X, Y, and Z represent the red, green, and blue densities of the reproductions, respectively, and the x, y,
The path of an Earth satellite can be written aswhere distances are in miles. What type of curve is represented? [x x y 2.76 -1 2.81 X y = [7.76 x 107]
Referring to Exercise 43, find the matrix B + J and explain the meaning of its elements.Data from Exercises 43One serving of brand K breakfast cereal provides the given percentages of the given
Solve the given systems of equations by Gaussian elimination.Exercise 38Data from Exercises 38Solve the given systems of equations using the inverse of the coefficient matrix. 3x + 2y + z = 22x
Solve the given systems of equations by Gaussian elimination.2x + 3y − z = 10x − 2y + 6z = −65x + 4y + 4z = 14
Using Kirchhoff’s laws on the circuit shown in Fig. 16.6, the following matrix equation is found. By matrix multiplication, find the resulting system of equations. R₁ + R₂ -R₂ 0 -R₂ R₂ +
Solve the given systems of equations by Gaussian elimination.x − 3y + 4z − 2t = 62x + y − 2z + 3t = 73x − 9y +12z− 6t = 12
Solve the systems of equations by determinants. As indicated, use the systems from Exercises.Exercise 35Data from Exercises 35Solve the given systems of equations using the inverse of the coefficient
Solve the systems of equations by determinants. As indicated, use the systems from Exercises.Exercise 36Data from Exercises 36Solve the given systems of equations using the inverse of the coefficient
Solve the systems of equations by determinants. As indicated, use the systems from Exercises.Exercise 37Data from Exercises 37Solve the given systems of equations using the inverse of the coefficient
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.3x − 2y + z = 62x + 3z = 34x
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check. 4rs 8t 2u + 4v = -1 + 3r +
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.7n + p + 2r = 34n − 2p + 4r =
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check. 2v + 3w + 2x - 2y + 5z =
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.2x − 3y + z − t = −84x +
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.6x + 4y − 4z − 4t = 05y +
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.3x − y + 6z − 2t = 82x + 5y
Solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.A + B + 2C − 3D = 15 3A
In Example 4, change the < to >, solve the resulting inequality, and graph the solution.Data from Example 4Solve the inequality |3 − 2x| < 3.We have the following solutionThe meaning of
Evaluate the given determinants by expansion by minors. 4-5 -1 16 4-3 6 -2
Use matrices A and B.Show that B4 = B. 10 4-[18] A 3 B 0 1 0 001 100
Evaluate the given determinants by expansion by minors. 7 2 1 -5 4 -6 3 0 -3
Use the determinants and evaluate each using a calculator. 4-5 -1 6 6 -2 1 4 -3
Evaluate the given determinants by expansion by minors. 1 4 0 3 1 2 -2 -2 -4 -1 6 -3 5 1 3-4

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