The answer to practice questions in Genki I and II
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MAT 111-18 MID-TERM EXAM ANSWER KEY For Test 1. MULTIPLE CHOICE. (1 POINT EACH ITEM) 1. E 2. B 3. C 4. B 5. D For Test Test 2. MATRICE MATRICESS MATHEMA MATHEMATIC TICAL AL OPERAT OPERATION IONSS & PROPERT PROPERTIES IES.. Perfor Perform m the indicat indicated ed mathem mathemati atical cal operations, if they exist. Show your complete solutions. Write your answer on provided spaces. 1.
Given the the following following matrices: matrices: (5 POINTS POINTS EACH OPERATION) OPERATION)
3 i 5 2 3i 5i
A
2 i 5 2i 3 i 4 3i
B
Complete the solution in finding the elements of matrix C if it is based on the following matrices operations:
c 22 9 1 15 ( 6 20 ) i c 22 23 26 i Answer:
6 19i 5 20i 14 13i 23 26i
C 2.
Solve Solve the following following system system of four four equations equations in four variable variabless by determi determinin ning g the invers inverse e of the matrix matrix of coefficients and then using matrix multiplication; AX = B or X = A -1B where all are matrices. (10 points).
x1 x2 2 x3 x4 5
2 x1 2 x3 x4 6 x2 3 x3 x4 1
3 x1 2 x2 2 x4 7 4 Page 1 Page 1 of of 4
Solution: Fill-up the elements of the given matrices based on given systems ofequations:
1 2 A 0 3
1
2
1
0
2
1
1 3 1 2 0 2
x1 5 x 6 2 B X x3 1 7 , x4 Solve for inverse of A, A -1 using Gauss - Jordan Elimination:
A : I 4 I 4 : A1 1 2 0 3
1
2
1
1
0
0
0
2
1
0
1
0
1
3
1 0 0 1
2
0
2
0
0
0
1 0 0 0 0 0 1 0
0
0
0
1
0
0
0
1
0
0
0
1
14
8
4
17 3
17 9
17 4
5 17 5
17 5
17 2
17 1
17 3
17 18
17 3
17 10
17 4
17
17
17
17
Obtained via 14 iterations of Gauss-Jordan Elimation as follows: 1) 5) 9) 13)
For Test 3. PROBLEM SOLVING. (20 points). John inherited $25,000 and invested part of it in a money market account, part in municipal bonds, and part in a mutual fund. After one year, he received a total of $1,620 in simple interest from the three investments. The money market paid 6% annually, the bonds paid 7% annually, and the mutually fund paid 8% annually. There was $6,000 more invested in the bonds than the mutual funds. Find the amount John invested in each category. Solution using Matrices and Determinants via Cramer’s Rule: Let x = investment in money market account y = investment in municipal bonds z = investment in mutual fund Equations as per problem: Eqn 1:
x + y + z = 25000
Eqn 2:
0.06x + 0.07y + 0.08z = 1620
Eqn 3:
z + 6000 = y
or re-arranging ===>
y - z = 6000
The augmented matrix representing system of linear equations is …
1 1 25000 1 0.06 0.07 0.08 1620 0 1 1 6000 Let A be the 3 x 3 sub-matrix of above matrix.
A (1)(0.07)( 1) (1)(0.08)(0) (1)(0.06)(1) (0)(0.07)(1) (1)(0.08)(1) ( 1)(0.06)(1) A 0.07 0 0.06 0 0.08 0.06 0.15 0.12 A 0.03 Using Cramer’s Rule, the values of x, y and z are determined:
