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The calculators most commonly used by Foundation Studies students at UTAR are Casio fx-350MS, fx-570MS, and fx-570ES. Casio fx-350MS has no matrix function and cannot be used to perform matrix calculation. Both Casio fx-570MS and Casio fx-570ES have matrix mode. Casio fx-570ES can display fully a 3×3 matrix with all its 9 elements visible. Casio fx-570MS has only a 2-line display and can only show matrix elements one at a time. Hence the operations for matrix computation are different for these two series of calculators. The general procedures in matrix calculations are as follows: (1) Enter Matrix mode of the calculators. (2) Assign a variable to store the matrix. There are 3 variables available: MatA, MatB, and MatC. (3) Select the dimension or order (1×1 to 3×3) of the matrix. (4) Input the elements of the matrix; the data will be automatically saved in the matrix variable assigned. (5) Exit the matrix input or edit mode by pressing the coloured AC key. (6) Press SHIFT MATRIX or SHIFT MAT (Matrix function is at numeric key 4) to recall the stored matrix and perform matrix calculations as needed.

(a) Casio fx-570ES

(a) Entering a matrix. 1. 2. 3. 4. Press the MODE key. Select 6:Matrix mode. A display Matrix? will be shown to let us select one of the 3 possible matrix variables allowable. Let us choose 1:MatA by pressing 1. The next display permits us to select the order (m×n) of the matrix MatA. Let us choose 1: 3×3 by pressing 1. An input screen will be displayed for us to key in the elements of the matrix. Let us key in the following matrix as an exercise:

1 1 2 4 0 3 5 1 2

Press 1=, 1=, 2=, 4=, 0=, 3=, 5=, 1=, and 2=. (The = key acts like the Enter key on a computer keyboard.) 5. Exit from the matrix input screen by pressing the AC key. (The AC acts like the ESC key on a computer keyboard.)

(b) Finding the inverse of a non-singular 3×3 matrix. 6. Next let us compute the inverse of MatA in memory that we have just keyed in. Press SHIFT MATRIX (Matrix function is at numeric key 4). Select option 3:MatA.

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7. 8.

The word MatA will be displayed. To find out its inverse, press the x-1 key. The display becomes MatA-1. Press the = key. After a short delay, the inverse of the matrix will be displayed:

1 − 4 7 12 1 3 0 − 2 3 1 3 1 4 5 12 1 − 3

(There will ne no inverse if the matrix is singular, i.e. its determinant is zero.) 9. The result of this matrix calculation, i.e. the inverse of the matrix, will be stored in the memory variable MatAns. We can do matrix calculation with MatAns. For example, we can multiply MatA with MatAns to get the identity matrix. Press SHIFT MATRIX. Select option 6:MatAns. MatAns will appear on the screen. Press the multiplication key ×. Press SHIFT MATRIX. Select 3: MatA. MatAns×MatA will appear on the screen. Press = to display the identity matrix:

1 0 0 0 1 0 0 0 1

10. 11.

(c) Finding the determinant of a 3×3 matrix. 12. 13. 14. We can also compute the determinant of matrix MatA. Press SHIFT MATRIX. Select 7:det. The screen will display det(. Select the matrix whose determinant is to be determined. Press SHIFT MATRIX. Choose 3:MatA by pressing 3. The display becomes det(MatA. Close the bracket by pressing ). Press = to display the determinant. The determinant of value 12 will be displayed.

(d) Finding the transpose of a matrix. 15. 16. 17. We can also display the transpose of the matrix easily. Press SHIFT MATRIX. Choose 8: Trn. The screen will display Trn(. Tell the calculator the matrix we want to transpose. Press SHIFT MATRIX. Choose 3: MatA by pressing key 3. Close the bracket by pressing ), followed by =. The transpose of MatA will be displayed.

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(e) Modifying the content of a matrix. 18. 19. To modify the matrix MatA already in memory, press SHIFT MATRIX. Choose 2:Data. The screen will display Matrix? 1: MatA 2: MatB 3: MatC. Since our matrix is stored in MatA, choose 1: MatA by pressing 1. The matrix A will be displayed. We can alter the values of the elements now. The changed values will be stored automatically when we exit from the edit display. Exit from the matrix edit screen by pressing the AC key.

20. 21.

(f) Multiplying one matrix by another matrix. 22. 23. 24. To multiply MatA already in memory by another memory, assign a variable to the second matrix first. Press SHIFT MATRIX. Select 2: Data. The screen will display Matrix? 2: MatB 3: MatC. 1: MatA Let us assign the second matrix to variable MatB. Select 2: MatB by pressing 2. The input screen for MatB will be displayed. As an exercise, enter the following matrix data:

2 3 1 5 7 3 4 0 6

25.

26. 27. 28.

Exit from the matrix edit screen by pressing the AC key. Now we can call MatA and MatB and multiply them together. Press SHIFT MATRIX. Select 3: MatA. MatA will appear on the screen. Press the multiplication key ×. Press SHIFT MATRIX. Select 4: MatB. MatA×MatB will appear on the screen. Press = to display the result:

15 10 16 20 12 22 23 22 20

(g) Solving system of linear equations in three variables. 29. Suppose we are required to solve the following system of linear equations: 2 x + 3 y + 3z = 0 4x − y − 6z = 1 − 4x + 3y + 6z = 3 3/13

The equations can be expressed in matrix form as: 2 3 3 x 0 4 −1 −6 y =−1 4 3 6 z 3 The values of x, y, z are given by −1 x 2 3 3 0 y = 4 −1 −6 −1 z 4 3 6 3 30. We now enter the values of the matrices. Press SHIFT MATRIX. Select 2: Data. Assign the first matrix to variable MatA and select the order of the matrix MatA as 3×3. Note: If MatA is already in memory, selecting 2: Data would bring up the already defined MatA and we can edit the matrix elements. If MatA is not yet in memory, then we need to assign MatA to the new matrix by selecting the order of the matrix. If MatA is already in memory and its order is not what we wanted, then we need to start from scratch - press the MODE key, select 6:Matrix mode, choose 1:MatA, and define the order. 31. An input screen will be displayed for us to key in the elements of the MatA:

2 3 3 4 −1 −6 4 3 6

32. 33. 34.

Exit from the matrix input screen by pressing the AC key. Next we create a second matrix. Press SHIFT MATRIX. Select 2: Data. Assign the second matrix to variable MatB and select the order of MatB as 3×1. The input screen for MatB will be displayed. Enter the following matrix data:

0 −1 3

35. 36.

Exit from the matrix edit screen by pressing the AC key. Now we are ready to find the solutions to the system of linear equations. Compute the inverse of MatA in memory that we have keyed in. Press SHIFT MATRIX. Select option 3:MatA. The word MatA will be displayed. To determine its inverse, press the x-1 key. The display becomes MatA-1. Press the = key. After a short delay, the inverse of MatA will be displayed: 4/13

37. 38.

1 5 1 − 6 8 24 1 2 0 − 3 3 7 − 2 − 1 12 36 9 The inverse matrix will be stored in memory as MatAns.

39. 40. 41.

Exit from the matrix edit screen by pressing the AC key. Recall from memory the inverse matrix we have just computed. Press SHIFT MATRIX. Select 6: MatAns. MatAns will appear on the screen. Press the multiplication key ×. Press SHIFT MATRIX. Select 4: MatB. MatAns×MatB will appear on the screen. Press = to display the result: 1 2 −1 2 3 x y = z 1 2 −1 2 3

42.

Hence,

Note: (i)

The defined matrix in memory can be cleared by changing the entry mode from matrix back to general computation: Press the MODE key. Select 1:COMP. Please note that using SHIFT CLR (key 9) does not clear the memory of the matrix entered.

(ii)

If seeing the data on the display is a problem, change the contrast in display using the following steps: Press Shift Setup, followed by the navigation key to display the second screen. Choose 6: Cont. Press the keys [Light] and [Dark] to change the display contrast as desired. Press AC to exit contrast adjustment.

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(b) Casio fx-570MS

(a) Entering a matrix. 1. Press the MODE key three times. The displays are as follows: First time : COMP CMPLX Second time : SD REG BASE Third time : EQN MAT VCT 1 2 3 2. Press 2 to select Matrix mode. A tiny MAT indicator will be visible at the central topmost row of the display. 3. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 Press 1 to select the order (or dimension) of the matrix. Always define the dimension of the matrix first when creating a new matrix. 4. The next display will show A B C 1 2 3 A, B, C stands for three possible matrix variables: MatA, MatB, MatC. Choose MatA by pressing 1. 5. The next display will be MatA (m×n) m? Press 3 followed by = to define the row number of a 3×3 matrix. (The = key functions like the Enter key on a computer keyboard.) 6. The next display will be MatA (m×n) n? Press 3 followed by = to define the column number of a 3×3 matrix As an exercise, key in the matrix below.

1 1 2 4 0 3 5 1 2

7. The next display is MatA11. Key in 1 followed by = to define the element for row 1 and column 1. 8. The next display is MatA12. Key in 1 followed by = to define the element for row 1 and column 2. 9. The next display is MatA13. Key in 2 followed by = to define the element for row 1 and column 3. 10. The next display is MatA21. Key in 4 followed by = to define the element for row 2 and column 1.

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11. The next display is MatA22. Key in 0 followed by = to define the element for row 2 and column 2. 12. The next display is MatA23. Key in 3 followed by = to define the element for row 2 and column 3. 13. The next display is MatA31. Key in 5 followed by = to define the element for row 3 and column 1. 14. The next display is MatA32. Key in 1 followed by = to define the element for row 3 and column 2. 15. The next display is MatA33. Key in 2 followed by = to define the element for row 3 and column 3. 16. This will bring us back to the beginning of the matrix MatA11. We may use the navigation key to scroll through all the elements for checking purpose. Press AC to exit matrix input mode. (The AC key functions like the ESC key on a computer keyboard – it clears the display screen.) (b) Finding the inverse of a non-singular 3×3 matrix. 17. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 Press 3 to select the 3×3 matrix just entered that is stored in memory. 18. The next display will be A B C 1 2 3 A, B, C stands for three matrices: MatA, MatB, MatC. Choose MatA by pressing 1. 19. The word MatA will be displayed. To find out its inverse, press the x-1 key. The display becomes MatA-1. Press = to find the inverse of the matrix, which is stored in variable MatAns. 20. The display becomes MatAns11, the row 1, column 1 of the inverse matrix element, the value of which is displayed at the second row of the screen. By default, the values b/c are in decimal form. Press the a key to change it into a fraction, if necessary. 21. Use the navigation key to move to the next column. The display is MatAns12 with the value at the second row of the screen. 22. Repeat pressing the navigation keys to display all the elements of the inverse matrix, an element at a time.

1 − 4 7 12 1 3 0 − 2 3 1 3 1 4 5 12 1 − 3

7/13

(There will ne no inverse if the matrix is singular, i.e. its determinant is zero.) 23. Press AC to exit from the matrix display mode. 24. The result of this matrix calculation, i.e. the inverse of the matrix, will be stored in the memory variable MatAns. We can do matrix calculation with MatAns. For example, we can multiply MatA with MatAns to get the identity matrix. 25. Press SHIFT MAT. Press 3 to select a matrix in memory. 26. The next display will be A B C Ans 1 2 3 4 The symbols stands for the four possible matrices: MatA, MatB, MatC, and MatAns. Choose MatAns by pressing 4. MatAns will appear on the screen. 27. Press the multiplication key ×. Press SHIFT MAT. Press 3 to select a matrix. 28. Press 1 to select MatA. MatAns×MatA will appear on the screen. Press = to display the identity matrix, again one element at a time.

1 0 0 0 1 0 0 0 1

(c) Finding the determinant of a 3×3 matrix. 29. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 30. Use the navigation key to scroll to the right: Det Trn 1 2 Press 1 to select Det. 31. The display will next show Det _ . Load the stored MatA by pressing SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Press 3 to call the 3×3 matrix just entered. 32. The next display will show A B C Ans 1 2 3 4 A, B, C stands for three possible matrices: MatA, MatB, and MatC. Choose MatA by pressing 1. 33. The display will become Det MatA. Press =. The value of the determinant will be displayed at the second screen row.

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(d) Finding the transpose of a matrix. 34. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 35. Use the navigation key to scroll to the next screen: Det Trn 1 2 Press 2 to select Trn. 36. The screen will display Trn _. Press SHIFT MAT. 37. The screen will display : Dim Edit 1 2 Press 3 to select a matrix. Mat 3

38. The next display will show A B C Ans 1 2 3 4 A, B, C stands for three possible matrices: MatA, MatB, and MatC. Choose MatA by pressing 1. The screen will display Trn MatA. Press = key. 39. The first element of the transpose of MatA MatAns11 will be displayed. Use the navigation keys or the = key to show the required element of the transposed matrix. (e) Modifying the content of a matrix. 40. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Edit by pressing 2. 41. The next display will show A B C Ans 1 2 3 4 A, B, C stands for three possible matrices: MatA, MatB, and MatC. Choose MatA by pressing 1. 42. The first element of MatA MatA11will be displayed. Change the value of the element as required. We may use the navigation keys to move to the required element. (f) Multiplying one matrix by another matrix. 43. To multiply MatA already in memory by another memory, assign a variable to the second matrix by first defining its dimension.

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44. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 Select Dim by pressing 1 to define the order of the second matrix. 45. The next display will show A B C Ans 1 2 3 4 Select B to assign the second matrix to MatB. Press 2. 46. The next display will be MatB (m×n) m? Press 3 followed by = to define the row number of a 3×3 matrix. The key = functions like the Enter key on a computer keyboard. 47. The next display will be MatB (m×n) n? Press 3 followed by = to define the column number of a 3×3 matrix. As an exercise, key in the following second matrix:

2 3 1 5 7 3 4 0 6

48. The next display is MatB11. Key in 2 followed by = to define the element for row 1 and column 1 of MatB. 49. The next display is MatB12. Key in 3 followed by = to define the element for row 1 and column 2. 50. Similarly key in all the remaining elements of MatB : MatB13, MatB21, MatB22, MatB23, MatB31, MatB32, and MatB33. 51. This will bring us back to the beginning of the matrix MatB11. Exit from the matrix input mode by pressing AC. 52. Now we can call MatA and MatB and multiply them together. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Mat by pressing 3. 53. The next display will show A B C 1 2 3 Select A to load in the matrix MatA. Press 1. Ans 4

54. MatA will appear on the screen. Press the multiplication key ×.

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55. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Mat by pressing 3. 56. The next display will show A B C 1 2 3 Select B to load in the matrix MatB. Press 2. 57. MatA×MatB will appear on the screen. Press =. 58. The first element of the matrix product MatAns11 will be displayed. Use the navigation keys to display the rest of the elements: MatAns12, MatAns13, MatAns21 MatAns22, MatAns23, MatAns31, MatAns32, and MatAns33. The resulting matrix should be:

15 10 16 20 12 22 23 22 20

Ans 4

(g) Solving system of linear equations in three variables. 59. Suppose we are required to solve the following system of linear equations: 2 x + 3 y + 3z = 0 4x − y − 6z = 1 − 4x + 3y + 6z = 3 The equations can be expressed in matrix form as: 2 3 3 x 0 4 −1 −6 y =−1 4 3 6 z 3 The values of x, y, z are given by

x y = z

2 3 3 0 4 −1 −6 −1 4 3 6 3

−1

60. We now enter the values of the matrices. Press SHIFT MATRIX. The screen will display : Dim Edit Mat 1 2 3 Select Edit by pressing 2. 61. The next display will show A B 1 2 Choose MatA by pressing 1. C 3 Ans 4

11/13

62. Since we have previously defined MatA, the first element of MatA MatA11 will be displayed. Replace MatA11 with 2. Press the navigation keys to move to MatA12 and replace it with 3. Continue using navigation keys to modify the matrix MatA as

2 3 3 4 −1 −6 4 3 6

63. Exit from the matrix input screen by pressing the AC key. 64. Next we create a second matrix. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Dim by pressing 1 to define the order of the new matrix. 65. The next display will show A B C Ans 1 2 3 4 Select B to assign the second matrix to MatB. Press 2. 66. The next display will be MatB (m×n) m? Press 3 followed by = to define the row number of a 3×1 matrix. 67. The next display will be MatB (m×n) n? Press 1 followed by = to define the column number of a 3×1 matrix. 68. The next display is MatB11. Enter 0 for MatB11, −1 for MatB21, 3 for MatB31 as in

0 −1 3

69. Exit from the matrix edit screen by pressing the AC key. 70. Now we are ready to find the solutions to the system of linear equations. Compute the inverse of MatA that we have keyed in. Press SHIFT MATRIX. Select option 3:Mat., and then 1: A. 71. The word MatA will be displayed. To determine its inverse, press the x-1 key. The display becomes MatA-1. 72. Press the multiplication key ×. Press SHIFT MATRIX. Select 3: Mat. , then select matrix 2: B. The first element of MatA-1×MatB, MatAns11, will appear on the screen. Use the navigation keys to display the other elements. The resulting matrix b/c MatAns should be (the results may be in decimal values; press the a key to change it into a fraction, if necessary):

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1 2 −1 2 3 x y = z 1 2 −1 2 3

73. Hence,

Note: The defined matrix in memory will be cleared by pressing the Shift CLR key.

Leong Sow Chew Lecturer Centre of Foundation Studies, Universiti Tunku Abdul Rahman (Kampar Campus)

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...Writing #4: Carbon Calculator Many of our daily activities trigger emissions of greenhouse gases. We produce greenhouse gas emissions by burning gasoline on our daily commutes, burning gas for home heating, or using electricity. Greenhouse gas emissions vary among individuals depending on a person's location, habits, and personal choices. It is good to know how much green house gasses an individual produces in their daily habits to be aware of their carbon footprint. From there a person can change their habits in an effort to reduce their footprint for a better life style and for the general health of the world. For this assignment I went through and checked all of my homes shared bills for accurate dollar amounts and measurements. For the questions about my car and my grandparent’s car, those were estimated through various websites including fuleeconomy.com. The lowest reliable value that I entered was my cars MPG. I found it easier to determine the average MPG from the reports on fuleeconomy.com rather than determining it myself. By the end of the carbon calculator, my estimated household carbon usage was 37,393 pounds of CO2 per year. Per individual this amount translates to roughly 12,464 pounds of CO2 per year. In comparison to the class average as seen below in figure 1.1, my house hold emissions were definitely on the lower end of the spectrum. To find the class average I took the higher number in the ranges offered, multiples each of those numbers by the responses......

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...FINANCIAL ANALYST Texas BA II Calculator Workshop CHARTERED FINANCIAL ANALYST Setting up your BAII Calculator Workshop Setting up your calculator (BAII Plus) Decimal places &|F! Set to mathematical precedence &|"&! No. of payments per year &-K Clear time value calculations &0 Calculator Workshop Memory function The calculator can store numbers for you Example: You calculate the answer to 2 + 3.5 = 5.5 and then wish to store it Press D then K (5.5 has now been stored and assigned to button K Having cleared the screen (P), it is now possible to recall the number by pressing J then K It is always possible to recall the last answer from the calculator by pressing & and then x CHARTERED FINANCIAL ANALYST Nominal vs. Effective Interest Rates Calculator Workshop Calculating nominal and effective rates Periodic rate (r) Nominal/stated rate r x Annual no. of periods Effective Annual Rate (EAR) EAR = (1 + r)12/n – 1 Where n = number of months Example: Calculating nominal and effective rates Calculate the nominal rate from a 4% six-monthly periodic rate Calculate the EAR from a 4% six-monthly rate Calculate the EAR from a nominal rate of 8% paid quarterly Calculator Workshop Example: 8% paid 6-monthly is, in effect, 8.16% 8% paid quarterly is, in effect, 8.243% 0.08 FV = $100 1 + = $108.243 4 r Effective rate = 1 + − 1 n where r is the nominal rate n 4 Calculator Workshop You......

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...and government and public services operate. In order to allow citizens to achieve a lifestyle that fits within one planet, governments need to dramatically improve the efficiency of the built environment and invest in renewable energy and smart land-use planning. (Network, 2014) In 2000, Wackemagel et al developed an Ecological Footprint Calculator that allows individuals assess their personal and societal impacts on the environment. It is critical for people involved in built environment and construction industry to have a broad understanding of sustainability and elements that have to be carefully considered to minimise the negative impacts on the surroundings we live in. In this report, I am going to use above calculator to evaluate my household ecological footprint located at 14 Waiben Crescent, Point Cook Vic 3030 and assess alternatives that can reduce the household Ecological Footprint. Below table is illustrates the findings of this calculation in 6 different categories based on the current lifestyle in September 2014. The Ecological Footprint per household member (presented as a land-use consumption matrix) FOSSIL ARABLE PASTURE FOREST CATEGORIES ENERGY LD. LAND 1.-FOOD 38131 126969 5960 0 2.-HOUSING 3747 0 0 3.-TRANSPORTATION 16657 0 4.-GOODS 1456 5.-SERVICES BUILT-UP expressed in average land with world average productivity......

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...swap can be calculated using standards methods of determining the present value of fixed leg and the floating leg. PV of fixed leg is computing as follows: , where C – fixed interest rate M – number of fixed payments P – notional ti – number of days in the period Ti – basis period dfi – discount factor Similarly, PV of floating leg: , where fi – floating rate And resulting swap present value: Excel work Vanilla interest rate swap calculator, developed and designed in Excel with some use of VBA, is completely based on swap theory described above. All the requirements imposed for the calculator are met. Before we value the plain vanilla swap, we need to input the following parameters: • Notional • Contract period (Since the majority of interest rate swaps have a contract period between 1 year and 5 years, I decided to specify it within this interval.) • Fixed interest rate • Frequency of payments (1, 2 or 4 times a year) After all the values of the required variables are entered, the button “Calculate” should be clicked to obtain the swap present value and such intermediate values like PV of fixed CF and PV of floating CF, the difference of which is actually the swap present value. The table below represents the logic of the swap calculator. Explanation of columns: • Column (1) represents maturity in parts of the year. The scale interval is set as 0.25, since maximum possible payment frequency (for this calculator) is 4 times......

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...Mortgage calculator One of my favorite tools and that I have used on many occasions is the mortgage amortization calculator from the internet. Using some basic information and a calculator it will help you have an elementary idea of monthly payments. Also, it can give you average information about the period of time needed for the full payment of the loan. Furthermore, an amortization schedule will show the payment, the interests, the distribution of capital, and the unpaid balance of the loan for each month until the full balance is paid off. The online mortgage calculator can be use easily without the need to understand complicated formulas. Simply enter the amount of the mortgage, interest rates and desire period of years, in order to be given immediately the amount of monthly payment plus a division of interest and the principal amount to be paid. With some calculator’s mortgage amortization, you can choose to check the consequences of the monthly over or lack of payments in efforts for the buyer to be mindful of the penalties if there are any in regards to their mortgage. Moreover, the debtor will determine how much they might save on interest payments if they were to pay the prematurely the loan. Prepared with awareness and understanding of your own financial abilities is key to make a tangible and wise decision on purchasing a home. Consequently you will be armed with the knowledge in efforts to negotiate terms and conditions that will be beneficial to your......

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...multiply the matrix by the position vector of the point. Example: A transformation is represented by the 2 by 2 matrix M = . To find the image of the point (3, 2) under this transformation, you need to find the result of the following matrix multiplication So the coordinates of the image are (8, -1). Example 2: A rectangle has coordinates (1, 1), (4, 1), (4, 3) and (1, 3). Find the coordinates of the image of the rectangle under the transformation represented by the matrix . Solution: You could find the image of each vertex in turn by finding , etc. However, it is more efficient to multiply the transformation matrix by a rectangular matrix containing the coordinates of each vertex: . So the image has coordinates (2, 0), (11, -3), (9, -1) and (0, 2). The diagram below shows the object and the image: Any transformation that can be represented by a 2 by 2 matrix, , is called a linear transformation. 1.1 Transforming the unit square The square with coordinates O(0, 0), I(1, 0), J(0, 1) and K(1, 1) is called the unit square. Suppose we consider the image of this square under a general linear transformation as represented by the matrix : . We therefore can notice the following things: * The origin O(0, 0) is mapped to itself; * The image of the point I(1, 0) is (a, c), i.e. the first column of the transformation matrix; * The image of the point J(0, 1) is (b, d), i.e. the second column of the transformation matrix; ......

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...Calculators, lemmings or frame-makers? The intermediary role of securities analysts Daniel Beunza and Raghu Garud Introduction As Wall Street specialists in valuation, sell-side securities analysts constitute a particularly important class of market actor.1 Analysts produce the reports, recommendations and price targets that professional investors utilize to inform their buy and sell decisions, which means that understanding analysts’ work can provide crucial insights on the determinants of value in the capital markets. Yet our knowledge of analysts is limited by insufﬁcient attention to Knightian uncertainty. Analysts estimate the value of stocks by calculating their net present value or by folding the future back into the present. In so doing, they are faced with the fundamental challenge identiﬁed by Frank Knight, that is, with the difﬁculty of making decisions that entail a future that is unknown. These decisions, as Knight wrote, are characterized by ‘neither entire ignorance nor complete . . . information, but partial knowledge’ of the world (Knight, [1921] 1971: 199). The ﬁnance literature has not examined the Knightian challenge faced by analysts. Indeed, existing treatments circumvent the problem by adopting one of two extreme positions. In the ﬁrst, put forward by orthodox economists, it is assumed that Knightian uncertainty is non-existent and that calculative decision-making is straightforward. Analysts are presented as mere calculators in a probabilistic......

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