Row 2 is equal to -2 times row 1, therefore according to property 5 the determinant is zero, and the matrix is singular.

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1 5 STK0 Assignment : Matri Algebra Part Section A: Homework Eercise H: Determinant of a Matri Do the following eercises on p78 of the tetbook: Memorandum H: No. b, f, 7a, 7c, 8e b. -(-6) - (-)(-) 0 6 f. 8 0 ()() + (0)() + (8)() - ()()() -(0)() - (8)() -0 7a. 6 8 Row is equal to - times row, therefore according to property 5 the determinant is zero, and the matri is singular. 7c All the elements of row are equal to zero, therefore according to property 6 the determinant is zero, and the matri is singular. 0 8e. A (coefficient matri is non-singular, therefore the system is solvable) A A 0 8 A

2 5 Eercise H: Application The supplier of television sets has shops in cities namely: Johannesburg, Cape Town and Durban. The sales of the television sets are summarized in the following table: TV-set City Type I Type II Type III Johannesburg 9 5 Cape Town Durban 5 The income (in R00) for each city, from the sales of these television sets, is as follows City Income (in R00) Johannesburg 7 Cape Town 68 Durban 70 Let: 9 5 S 8 6 0, N en E. Assume: The selling prices of each type of television set is the same for the cities S 00 en S Answer the following by making use of matri operations: a. Calculate the total number of televisions that the supplier has sold in each city. b. Calculate the selling price (in R00) for the different television sets. c. Calculate the total number of televisions that the supplier has sold from every type of television. d. Suppose that the sales in matri S doubled, then the matri of sales is given by 8 0 S 6 0. Determine the value of S 0 Memorandum H:. a. SE b. S N c. S E ( ) ( 7 0) d. S 00 S (Property of the determinant of a matri)

Figure : Formula Sheet Figure : Value Sheet Eercise P: Solving a system of linear equations by means of Cramer s Rule Solve the following system of linear equations by using Ecel.

3 55 Section B: Practical Eercise P: Determinant of a Matri a. Enter matri A into cells B:D. (Figure ) b. Calculate the determinant of matri A by entering the following formula into cell B6: mdeterm(b:d) c. Press Enter. Figure : Formula Sheet Figure : Value Sheet Eercise P: Solving a system of linear equations by means of Cramer s Rule Solve the following system of linear equations by using Ecel. + 7 Refer to the formula worksheet (Figure ) and value worksheet (Figure ) given below when performing the tasks. a. Enter the coefficient matri and calculate its determinant. Enter the elements of the coefficient matri A into cells B:D. Calculate the determinant of A by entering the following formula into cell B: MDETERM(B:D) Since A 0 a unique solution for, and can be obtained. b. Enter the vector with constant terms into cells B7:B9. c. Enter the matrices used in the numerators of the equations used to solve, and. The elements of the matri in the numerator of the equation used to solve are entered into cells F:H. This matri is the same as A with the first column replaced by b, b and b. The elements of the matri in the numerator of the equation used to solve are entered into cells F7:H9. This matri is the same as A with the second column replaced by b, b and b. Finally, the elements of the matri in the numerator of the equation used to solve are entered into cells F:H. This matri is the same as A with the third column replaced by b, b and b. d. Calculate the determinants of the matrices in step (c) in cells J, J7 and J respectively. e. Find the solution of the system of linear equations. The formulas for solving, and are entered into cells B7:B9

4 Figure : Formula worksheet 56 Figure : Value worksheet Eercise P: The Inverse of a Matri a. Enter Matri A into cells B:D. Select cells B7:D9, the cells in which we want the inverse to appear. This is shown in Figure 5. b. Type the following formula: minverse(b:d). DO NOT PRESS ENTER. This step is shown in Figure 6. c. Press CTRL + SHIFT + ENTER and the array formula will be entered into each of the cells B7:D9 The result is shown in Figure 7.

57 Figure 5 Figure 6 Figure 7 Eercise P: Solving a System of Linear Equations by means of the Inverse Matri Approach Solve the following system of linear equations by using Ecel.

5 57 Figure 5 Figure 6 Figure 7 Eercise P: Solving a System of Linear Equations by means of the Inverse Matri Approach Solve the following system of linear equations by using Ecel. 7 + where X A and 7 B Refer to the formula worksheet (Figure ) and value worksheet (Figure ) given below when performing the tasks. a. Enter the coefficient matri and calculate the inverse. Enter the matri into cells B:D. Select cells B7:D9 and enter the formula minverse(b:d). Press CTRL + SHIFT + ENTER. b. Enter the vector B with constant terms into cells B:B. c. Calculate the product B A. Selects cells B7:B9, the cells in which we want the product to appear. Type the following formula: mmult(b7:d9,b:b). Press CRTL + SHIFT + ENTER.

6 58 Figure 8: Formula Sheet Figure 9: Value Sheet Eercise P5: Solve the following system of linear equations by: a. Cramer s Rule b. Inverse Matri Approach r s + t 9 r t 7 r + t 5 Memorandum P5 r. s.89 t.67 Figure 0: Formula Sheet for Cramer s Rule

Figure : Formula Sheet for the Inverse Matri Approach 59 Section C: Typical Eam Questions Question The inverse matri of A is: Hint: Use the definition of an inverse matri.

7 Figure : Formula Sheet for the Inverse Matri Approach 59 Section C: Typical Eam Questions Question The inverse matri of A is: Hint: Use the definition of an inverse matri. (A) (C) (E) (B) (D) 0 0 Questions to are based on the following information: The following table contains the number of subscribers (in thousand) at three cellular service providers across three regions for August 00: Region Service provider X Y Z A 0 70 B 5 5 C 68 The regional total income (in Rand) from service fees is: Region Income (in R000) A 5 70 B C Let 0 70 N 5 5, R and ( ) E. Assume: The monthly service fee (in Rand) of the service providers is the same across the regions.

8 Question The total number of subscribers per region is: 60 (A) NR (C) E (E) (B) NE N (D) R N E N Question The monthly service fee (in Rand) of each service provider is: (A) NR (B) R N (C) N E (D) N R (E) E N Question The following information is available in Ecel: Formula worksheet: Value worksheet:

9 The monthly service fee (in Rand) charged by service provider X is: (A) 0 (B) 0 (C) (D) 97 (E) Question 5 Which one of the following matrices is non-singular? (A) (C) (E) (B) (D) Question 6 Let A be a ( m n) matri and B be a ( n) Which one of the following is not true. (A) ca ( ca ij ) for every i and j (B) Matri A will be square if (C) It is possible to calculate BA A for i... m a ij (D) ( ) (E) BI IB B m n and j... n n matri. Question 7 Which one of the following is not a characteristic of a determinant of a matri? (A) A A' (B) Determinants are defined only for square matrices (C) The interchanging of any two rows (or any two columns) will alter the sign, but not the numerical value, of the determinant (D) If a row (or column) of a matri contains only zeros, then the value of the determinant is equal to zero (E) If one row (or column) is a multiple of another row (or column), the value of the determinant will be zero Memo: Q B, Q C, Q D, Q A, Q5 D Q6 C Q7 A

3. Replace any row by the sum of that row and a constant multiple of any other row.

3. Replace any row by the sum of that row and a constant multiple of any other row. Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or