Math 285 Quiz One Name: ________________ 1. The product of these two matrices 5 2 2 −1  4 −2 1 3  is (Please show your work for full credit.) 2. For what value of a is this determinant equal to 0? 0 5 2 0 1 −8 −4 2 (please justify your answer.) 3. What is the product of these matrices? 3 2 6 −2 1 0 4 1 0   2346  (Please justify your answer.) 4. What is the largest possible number of pivots a 7 × 5 matrix can have? (Please justify your answer.) Homework 1 5. Find the standard matrix of the linear transformation : →  which rotates a point about the origin through an angle of   radians (counterclockwise). True or False (Simply enter T or F, no need to justify the answer) If matrices  and  are row equivalent, they have the same reduced echelon form. In general,  +  ≠  +  (assume  and  are  x  matrices). If a matrix  is symmetric, then so is  + “#. A matrix  must be a square matrix to be invertible. If $%&’( ≠ 0, then columns of  are linearly independent. If an  x  matrix  is equivalent to “#, then )* is also equivalent to “#. If an + x  matrix  has a pivot position in every row, then the equation , = . has a unique solution for each . in /0. If  = “, then  is invertible.

Math 285 Quiz One Name: ________________ 1. The product of these two matrices 5 2 2 −1  4 −2 1 3  is (Please show your work for full credit.) 2. For what value of a is this determinant equal to 0? 0 5 2 0 1 −8 −4 2 (please justify your answer.) 3. What is the product of these matrices? 3 2 6 −2 1 0 4 1 0   2346  (Please justify your answer.) 4. What is the largest possible number of pivots a 7 × 5 matrix can have? (Please justify your answer.) Homework 1 5. Find the standard matrix of the linear transformation : →  which rotates a point about the origin through an angle of   radians (counterclockwise). True or False (Simply enter T or F, no need to justify the answer) If matrices  and  are row equivalent, they have the same reduced echelon form. In general,  +  ≠  +  (assume  and  are  x  matrices). If a matrix  is symmetric, then so is  + “#. A matrix  must be a square matrix to be invertible. If $%&’( ≠ 0, then columns of  are linearly independent. If an  x  matrix  is equivalent to “#, then )* is also equivalent to “#. If an + x  matrix  has a pivot position in every row, then the equation , = . has a unique solution for each . in /0. If  = “, then  is invertible.

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Suppose a force analysis of a particle in equilibrium produced the following system of equations: 4F C ! 3F B = 500 0.866F A + 0.707F B ! F C = 0 !0.707F B + 0.5F A = 50 Solve this system of linear equations using the Gauss-Jordan Elimination method. a) Show the augmented matrix b) Use elementary row operations to obtain the reduced row echelon form c) Write the solution for the three forces (using 3 significant digits), FA, FB, & FC

Suppose a force analysis of a particle in equilibrium produced the following system of equations: 4F C ! 3F B = 500 0.866F A + 0.707F B ! F C = 0 !0.707F B + 0.5F A = 50 Solve this system of linear equations using the Gauss-Jordan Elimination method. a) Show the augmented matrix b) Use elementary row operations to obtain the reduced row echelon form c) Write the solution for the three forces (using 3 significant digits), FA, FB, & FC

 
Essential Statistics for Public Managers and Policy Analysts / Edition 3 by Evan M Berman, Xiaohu Wang 1-Use the public perception dataset. Is the relationship between watching Orange TV (watch), the county’s cable television station, and trusting the government to do what is right most of the time (trust) statistically significant? Do you consider this a causal relationship or an association? Does the analysis satisfy the assumptions of the Chi-square test? If not, how might you address this problem? 2-Use the public perception dataset. Examine the relationship between residents who trust the county government to do what is right most of the time (trust) and their belief that county government works efficiently (works). What is the practical significant of this relationship? 3-Use the public perception dataset. In Chapter 10 of this workbook, you used Chi-square to examine the relationship between residents who trust the county government to do what is right most of the time (trust) and their belief that county government works efficiently (works). Reexamine this relationship using measures of gamma, Somer’s d, Kendall’s tau-c. What do you conclude? 4-Table W 12.1 is the printout of a t-test (independent samples). The continuous variable is an index variable of environmental concern. The dichotomous variable is a measure of education (college versus no college). Interpret and write up the results. What other information would you like to have about this relationship? 5-Table W 12.2 is the printout of a period-samples t-test. The data are before-and-after measurements of a public safety program. Interpret and write up the results. What other information would you like to have about this relationship? 6-Use the Public Perception dataset. An analyst wants to know whether incomes vary by age group. Treat the income variable as a continuous variable, and treat the age variable as an ordinal variable. Calculate the means for each of these groups, and then use ANOVA to determine whether any of these differences are statistically significant. For which group is the relationship linear?

Essential Statistics for Public Managers and Policy Analysts / Edition 3 by Evan M Berman, Xiaohu Wang 1-Use the public perception dataset. Is the relationship between watching Orange TV (watch), the county’s cable television station, and trusting the government to do what is right most of the time (trust) statistically significant? Do you consider this a causal relationship or an association? Does the analysis satisfy the assumptions of the Chi-square test? If not, how might you address this problem? 2-Use the public perception dataset. Examine the relationship between residents who trust the county government to do what is right most of the time (trust) and their belief that county government works efficiently (works). What is the practical significant of this relationship? 3-Use the public perception dataset. In Chapter 10 of this workbook, you used Chi-square to examine the relationship between residents who trust the county government to do what is right most of the time (trust) and their belief that county government works efficiently (works). Reexamine this relationship using measures of gamma, Somer’s d, Kendall’s tau-c. What do you conclude? 4-Table W 12.1 is the printout of a t-test (independent samples). The continuous variable is an index variable of environmental concern. The dichotomous variable is a measure of education (college versus no college). Interpret and write up the results. What other information would you like to have about this relationship? 5-Table W 12.2 is the printout of a period-samples t-test. The data are before-and-after measurements of a public safety program. Interpret and write up the results. What other information would you like to have about this relationship? 6-Use the Public Perception dataset. An analyst wants to know whether incomes vary by age group. Treat the income variable as a continuous variable, and treat the age variable as an ordinal variable. Calculate the means for each of these groups, and then use ANOVA to determine whether any of these differences are statistically significant. For which group is the relationship linear?

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Vermont Technical College Electronic Applications ELT-2060 Lab 05: DC characteristics, input offset voltage and input bias current Reference: Operational Amplifiers with Linear Integrated Circuits Fourth edition William D. Stanley, pages 154-155 (Problems 3-21, 3-22 and Lab exercises LE 3-1 to LE 3-4) For the following exercises, make sure to record all calculations, estimations and measured results. Components: 2 741 Op Amps, 10k Ω Potentiometer, 4-10kΩ, 1kΩ , 100kΩ , 100Ω , 560kΩ , 5.6M Ω, resistors Objectives: a. Voltage offset Null Circuit and Closed-loop Differential Circuit b. Measurement of dc Input Offset Voltage c. Measurement of dc Bias and Offset Currents a. Voltage offset Null Circuit and Closed-loop Differential Circuit In this exercise, investigate the use of a null circuit to reduce the output dc offset to its minimum possible value. Refer to the “Voltage Offset Null Circuit” describe in the 741 op amp data sheet from Appendix C of your text book. Although there are no specific closed-loop configurations shown, use a closed-loop differential Amplifier shown in Figure 1. The differential nature of this type of circuit makes it particularly sensitive, therefore well suited, to illustrate the concept dc voltage offset. 1. Connect the closed-loop difference amplifier of Figure 1 with R=10k Ω and A=1. Using a 10kΩ potentiometer connect the “Voltage Offset Null Circuit” between nodes 1 and 5 as shown in the 741 data sheet. Keep in mind that a potentiometer is a three terminal device. You will need to connect the potentiometer wiper terminal to the lowest potential in the circuit -VCC. 2. Connect the two external circuit inputs (v1 and v2) to ground, measure the dc voltage. From the data sheet the expected value of offset voltage at room temperature is 2mV typical and 6mV maximum. Voltages at these levels will be hard to measure with the laboratory multimeter. 3. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 5. Do not break down you difference amplifier. Next, build the non-inverting amplifier as shown in figure 2 with Ri=1k Ω and Rf =100k Ω. Attach the output of the difference amplifier to the input of the non-inverting amplifier. This will amplify your offset by 101. 6. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 7. In effect we amplified the voltage offset from the difference amplifier by 101. Please describe any possible flaws in using this approach. Compare this result to what was measured in step 2. 8. Write an equation that expresses the expected output voltage Vo in terms of the two input voltages V1 and V2. 9. Apply dc input voltage for the following six combinations, compare the results to the expected values you calculate with the equation from step 8 a. V1=10V, V2=0V b. V1=0V, V2=10V c. V1=V2=10V d. V1=10mV, V2=0 e. V1=0, V2=10mV f. V1=V2=10mV b. Measurement of dc Input Offset Voltage ( Stanley Problem 3-21 page 151) A circuit and equation to measure the input offset voltage Vio is show in figure 3. With the proper selection of resistors Ri, Rf, and Rc the effects of offset due to input bias currents can be neglected. When the input terminals are both held to ground the resulting output voltage should be a direct measurement of Vio. 1. Build the circuit in Figure 3 with Ri=100 Ω and Rf=10k Ω measure and record Vo. Compare your results with the specification of input offset voltage provided in the data sheet. 2. Increase the value of Rf to 100k Ω, and measure Vo again. Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input bias currents are negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on how there relationship demonstrates that neglecting input bias current was a valid assumption. c. Measurement of dc Bias and Offset Currents (Stanley Problem 3-22 page 152) Consider the three circuits of figure 4 .The resistance R is chosen large so that the contribution to the output from bias currents is considerably larger than the contribution from the input offset voltages. The accompanying equations will predict the values of Ib+, Ib- and Iio. 1. Start with setting R=560k Ω and build each circuit in figure 4 one at a time. Going from one configuration to the next configuration should be quick, all that is changing is the placement of the resistors. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet. 2. Increase the value of R to 5.6M Ω. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet and to the results in part 1.Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input offset voltage effect is negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on why neglecting input offset voltage was a valid assumption. LAB write up: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. No MultiSim will be required for this report. Please include a written English language paragraph for all lab steps that required an explanation.

Vermont Technical College Electronic Applications ELT-2060 Lab 05: DC characteristics, input offset voltage and input bias current Reference: Operational Amplifiers with Linear Integrated Circuits Fourth edition William D. Stanley, pages 154-155 (Problems 3-21, 3-22 and Lab exercises LE 3-1 to LE 3-4) For the following exercises, make sure to record all calculations, estimations and measured results. Components: 2 741 Op Amps, 10k Ω Potentiometer, 4-10kΩ, 1kΩ , 100kΩ , 100Ω , 560kΩ , 5.6M Ω, resistors Objectives: a. Voltage offset Null Circuit and Closed-loop Differential Circuit b. Measurement of dc Input Offset Voltage c. Measurement of dc Bias and Offset Currents a. Voltage offset Null Circuit and Closed-loop Differential Circuit In this exercise, investigate the use of a null circuit to reduce the output dc offset to its minimum possible value. Refer to the “Voltage Offset Null Circuit” describe in the 741 op amp data sheet from Appendix C of your text book. Although there are no specific closed-loop configurations shown, use a closed-loop differential Amplifier shown in Figure 1. The differential nature of this type of circuit makes it particularly sensitive, therefore well suited, to illustrate the concept dc voltage offset. 1. Connect the closed-loop difference amplifier of Figure 1 with R=10k Ω and A=1. Using a 10kΩ potentiometer connect the “Voltage Offset Null Circuit” between nodes 1 and 5 as shown in the 741 data sheet. Keep in mind that a potentiometer is a three terminal device. You will need to connect the potentiometer wiper terminal to the lowest potential in the circuit -VCC. 2. Connect the two external circuit inputs (v1 and v2) to ground, measure the dc voltage. From the data sheet the expected value of offset voltage at room temperature is 2mV typical and 6mV maximum. Voltages at these levels will be hard to measure with the laboratory multimeter. 3. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 5. Do not break down you difference amplifier. Next, build the non-inverting amplifier as shown in figure 2 with Ri=1k Ω and Rf =100k Ω. Attach the output of the difference amplifier to the input of the non-inverting amplifier. This will amplify your offset by 101. 6. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 7. In effect we amplified the voltage offset from the difference amplifier by 101. Please describe any possible flaws in using this approach. Compare this result to what was measured in step 2. 8. Write an equation that expresses the expected output voltage Vo in terms of the two input voltages V1 and V2. 9. Apply dc input voltage for the following six combinations, compare the results to the expected values you calculate with the equation from step 8 a. V1=10V, V2=0V b. V1=0V, V2=10V c. V1=V2=10V d. V1=10mV, V2=0 e. V1=0, V2=10mV f. V1=V2=10mV b. Measurement of dc Input Offset Voltage ( Stanley Problem 3-21 page 151) A circuit and equation to measure the input offset voltage Vio is show in figure 3. With the proper selection of resistors Ri, Rf, and Rc the effects of offset due to input bias currents can be neglected. When the input terminals are both held to ground the resulting output voltage should be a direct measurement of Vio. 1. Build the circuit in Figure 3 with Ri=100 Ω and Rf=10k Ω measure and record Vo. Compare your results with the specification of input offset voltage provided in the data sheet. 2. Increase the value of Rf to 100k Ω, and measure Vo again. Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input bias currents are negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on how there relationship demonstrates that neglecting input bias current was a valid assumption. c. Measurement of dc Bias and Offset Currents (Stanley Problem 3-22 page 152) Consider the three circuits of figure 4 .The resistance R is chosen large so that the contribution to the output from bias currents is considerably larger than the contribution from the input offset voltages. The accompanying equations will predict the values of Ib+, Ib- and Iio. 1. Start with setting R=560k Ω and build each circuit in figure 4 one at a time. Going from one configuration to the next configuration should be quick, all that is changing is the placement of the resistors. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet. 2. Increase the value of R to 5.6M Ω. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet and to the results in part 1.Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input offset voltage effect is negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on why neglecting input offset voltage was a valid assumption. LAB write up: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. No MultiSim will be required for this report. Please include a written English language paragraph for all lab steps that required an explanation.

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A cosmetic X marketing department has developed a linear trend equation that can be used to predict annual sales of its popular Hand & Foot cream. Yt = 80,000 + 15t; where Yt = Annual sales; t = 0 corresponds to year 2000 Predict the annual sales for the year 2005 using the equation.

A cosmetic X marketing department has developed a linear trend equation that can be used to predict annual sales of its popular Hand & Foot cream. Yt = 80,000 + 15t; where Yt = Annual sales; t = 0 corresponds to year 2000 Predict the annual sales for the year 2005 using the equation.

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1. (20 pts) The linear momentum operator in one dimension is given by: ˆpx = ! i d dx . In class we said that the average momentum for a particle in a box (pib) is 0. Use the formula for the expectation value to verify mathematically that this is true. The pib-wavefunction: 2 ℓ sin nπ ℓ x ! ” # $ % & 2. (20 pts) Evaluate the following commutators: a. (10 pts) b. (10 pts) 3. (20 pts) A certain one-dimensional quantum mechanical system is described by the Hamiltonian: , q is a constant, and 0 ≤ x ≤ ∞. One of the eigenfunctions is known to be: a. (15 pts) Find the value of N to normalize the function. b. (5 pts) By letting , find the energy eigenvalue. 4. (20 pts) The Schrödinger equation for the particle on a sphere (a.k.a. the Rigid Rotor) is: − !2 2μR2 1 sinθ ∂ ∂θ sinθ ∂ ∂θ # $ % & ‘ ( + 1 sin2θ ∂2 ∂φ 2 # $ % & ‘ ( ψ(θ,φ ) = Eψ(θ,φ ) A purported eigenfunction for it is: ψ(θ,φ ) = N sin3θ cos(3φ ) a. (15 pts) Use this wave function to find the energy eigenvalue for the function. (You do NOT have to normalize the function!). b. (5 pts) The eigenvalues for the particle on a sphere are of the form: Eℓ = “2 2μR2 ℓ(ℓ +1) What is the value of ℓ for the wave function used in part a? 5. (20 pts) Using the ortho-normailty of the hydrogenic orbitals and the spin functions, normalize the excited Helium atom represented by the following wave function: ψ = N 1s({ 1)2p(2)+ 2p(1)1s(2)}{α(1)β (2)−β (1)α(2)} ˆ x, ( ˆpx [ + xˆ)] = ˆpx, ( ˆ x)3 !” #$ = ˆH = − 2 2m d2 dx2 − q2 x ψ(x) = Nxe−α x α = mq2 / 2

1. (20 pts) The linear momentum operator in one dimension is given by: ˆpx = ! i d dx . In class we said that the average momentum for a particle in a box (pib) is 0. Use the formula for the expectation value to verify mathematically that this is true. The pib-wavefunction: 2 ℓ sin nπ ℓ x ! ” # $ % & 2. (20 pts) Evaluate the following commutators: a. (10 pts) b. (10 pts) 3. (20 pts) A certain one-dimensional quantum mechanical system is described by the Hamiltonian: , q is a constant, and 0 ≤ x ≤ ∞. One of the eigenfunctions is known to be: a. (15 pts) Find the value of N to normalize the function. b. (5 pts) By letting , find the energy eigenvalue. 4. (20 pts) The Schrödinger equation for the particle on a sphere (a.k.a. the Rigid Rotor) is: − !2 2μR2 1 sinθ ∂ ∂θ sinθ ∂ ∂θ # $ % & ‘ ( + 1 sin2θ ∂2 ∂φ 2 # $ % & ‘ ( ψ(θ,φ ) = Eψ(θ,φ ) A purported eigenfunction for it is: ψ(θ,φ ) = N sin3θ cos(3φ ) a. (15 pts) Use this wave function to find the energy eigenvalue for the function. (You do NOT have to normalize the function!). b. (5 pts) The eigenvalues for the particle on a sphere are of the form: Eℓ = “2 2μR2 ℓ(ℓ +1) What is the value of ℓ for the wave function used in part a? 5. (20 pts) Using the ortho-normailty of the hydrogenic orbitals and the spin functions, normalize the excited Helium atom represented by the following wave function: ψ = N 1s({ 1)2p(2)+ 2p(1)1s(2)}{α(1)β (2)−β (1)α(2)} ˆ x, ( ˆpx [ + xˆ)] = ˆpx, ( ˆ x)3 !” #$ = ˆH = − 2 2m d2 dx2 − q2 x ψ(x) = Nxe−α x α = mq2 / 2

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Lectorial 5: The Gravitron The Gravitron (shown in figure 1 [1]) is a carnival ride designed to simulate the experience of zero gravity. The ride consists of a 15 metre diameter circular chamber which spins around a centre shaft. The spinning motion applies a force to the occupants of the ride pinning them up against their seat. Figure 1: The Gravitron carnival ride. For this lectorial task we want to study the forces being applied to the ride’s occupants and determine the g-forces they would be experiencing. According to physics, the rules for uniform circular motion are: where: 1. If the ride has a maximum rotational speed of 24 revolutions per minute (rpm), determine the force being applied to the ride’s occupants. What gforces are the people experiencing (assume occupants are 65 kg adults)? [1] “Gravitron” used under Creative Commons licence (https://creativecommons.org/licenses/by-nc-sa/2.0/). Photo by: bobdole369 Newtons 2 r v F = ma = m angular speed in radians per second rotational speed in revolutions per second (or Hz) radius of the Gravitron tangential velocity of the Gravitron mass of occupant = = = = = w f r v m -1 v = wr ms w = 2pf rad/sec Typically the Gravitron ride takes approximately 20 seconds to reach its maximum rotational speed of 24 rpms and the whole ride lasts for around 80 seconds. This means the ride’s occupants are exposed to non-uniform circular motion meaning there is changing linear velocity at certain parts of the ride. For non-uniform circular motion the following formulae are useful: where: A GPS tracking device was attached to a person in the Gravitron and data was obtained about their x,y displacement vs. time over the 80 second duration of the ride. The data was saved in a .csv file called ‘gravitron.csv.’ This file contains three columns: time, x-displacement and y-displacement, e.g.: Time, sec x-displacement y-displacement 0.00 0.10 0.20 … 2. Download this .csv file from Blackboard. Find the g-forces being applied to the ride’s occupants for the whole 80 second duration of the ride. Again assume the occupants are 65 kg adults. Think about how you could effectively present these results. -2 2 ms r v -1 a = 2 2 ms     +     = dt dy dt dx v centripetal acceleration time in seconds displacement in y direction displacement in x direction = = = = a t y x

Lectorial 5: The Gravitron The Gravitron (shown in figure 1 [1]) is a carnival ride designed to simulate the experience of zero gravity. The ride consists of a 15 metre diameter circular chamber which spins around a centre shaft. The spinning motion applies a force to the occupants of the ride pinning them up against their seat. Figure 1: The Gravitron carnival ride. For this lectorial task we want to study the forces being applied to the ride’s occupants and determine the g-forces they would be experiencing. According to physics, the rules for uniform circular motion are: where: 1. If the ride has a maximum rotational speed of 24 revolutions per minute (rpm), determine the force being applied to the ride’s occupants. What gforces are the people experiencing (assume occupants are 65 kg adults)? [1] “Gravitron” used under Creative Commons licence (https://creativecommons.org/licenses/by-nc-sa/2.0/). Photo by: bobdole369 Newtons 2 r v F = ma = m angular speed in radians per second rotational speed in revolutions per second (or Hz) radius of the Gravitron tangential velocity of the Gravitron mass of occupant = = = = = w f r v m -1 v = wr ms w = 2pf rad/sec Typically the Gravitron ride takes approximately 20 seconds to reach its maximum rotational speed of 24 rpms and the whole ride lasts for around 80 seconds. This means the ride’s occupants are exposed to non-uniform circular motion meaning there is changing linear velocity at certain parts of the ride. For non-uniform circular motion the following formulae are useful: where: A GPS tracking device was attached to a person in the Gravitron and data was obtained about their x,y displacement vs. time over the 80 second duration of the ride. The data was saved in a .csv file called ‘gravitron.csv.’ This file contains three columns: time, x-displacement and y-displacement, e.g.: Time, sec x-displacement y-displacement 0.00 0.10 0.20 … 2. Download this .csv file from Blackboard. Find the g-forces being applied to the ride’s occupants for the whole 80 second duration of the ride. Again assume the occupants are 65 kg adults. Think about how you could effectively present these results. -2 2 ms r v -1 a = 2 2 ms     +     = dt dy dt dx v centripetal acceleration time in seconds displacement in y direction displacement in x direction = = = = a t y x

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consider the following linear equations 2x+y-3z=-7, 3x+2y+z=11, 4x+3y-2z=1 a) write these equations in matrix form b) find the determine of the matrix of coefficient c) solve the equation using the matrix inversion method

consider the following linear equations 2x+y-3z=-7, 3x+2y+z=11, 4x+3y-2z=1 a) write these equations in matrix form b) find the determine of the matrix of coefficient c) solve the equation using the matrix inversion method