MAT 205 CUMULATIVE TEST 2 (CT2)
MAT 205
WEEK 4
CUMULATIVE TEST 2
(CT2)
1. What is the size of the matrix below? What is its additive inverse?
[pic]
|Show your work here. |
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|The size of the matrix is 2 x 3 |
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|The additive inverse is [pic] |
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2. The identity matrix I is defined below. Using the definition of matrix multiplication, and writing out the element multiplications, prove for any matrix A, also defined below, that AI=IA=A.
I=[pic] , A=[pic]
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|AI = [pic] |
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|[pic] |
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3. Two generalized square matrices A, B, are shown below. Is matrix multiplication for square matrices commutative? Prove your answer using the definition of matrix multiplication. Show a 2x2 example that supports your proof. Hint; consider one element of the multiplication.
A=[pic] , B=[pic]
|Show your work here. |
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|The first row first column element of AB is: |
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|a11b11 + a12b212 + a13b31 |
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|The first row first column element of BA is: |
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|b11a11 + b12a21 + b13a31 |
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|Clearly the two elements are not equal in a general sense, although there may be particular matrices for which the two elements |
|would be equal. |
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|2x2 Example: |
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|[pic] |
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4. Find the inverse of A using text to describe your row operations, and equation editor to show the row operation results. When you get the inverse, A-1, get its inverse, (A-1)-1 and show that it is the original matrix below.
A=[pic]
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|Inverse of A: |
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|[pic] |
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|Inverse of Inverse of A: |
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|[pic] |
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5. Does a 3x3 matrix with 1 row with elements all equal to 0 have an inverse? Explain your answer using the details of the augmented matrix approach to getting an inverse.
|Show your work here. |
|[pic] |
6. In the equations below, x1 and x2 are variables, and all the other quantities are constants. Put the equations in matrix form, and use matrix inversion to solve for x1 and x2.
a11x1+a12x2=c1
a21x1+a22x2=c2
|Show your work here. |
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|See next page. |
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|[pic] |
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7. Write the solutions that can be read from the following simplex maximization tableau.
x1 x2 x3 s1 s2 s3 z
[pic]
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|2z = 40 ( z = 20 |
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|7x1 = 35 ( x1 = 5 |
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|x3 = 2 |
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|5s1 = 15 ( s1 = 3 |
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|x2 = s2 = s3 = 0 |
8. Use slack variables as needed, write the initial simplex tableau, then find the solution and the maximum value.
Maximize z=2x1+3x2
Subject to: 3x1+5x2[pic]29
2x1+x2[pic]10
With: x1[pic]0 and x2[pic]0
|Show your work here. |
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|See next page. |
|[pic] |
9. State the dual problem for this linear programming problem.
Maximize z=8x1+3x2+x3
Subject to: 7x1+6x2+8x3[pic]18
4x1+5x2+10x3[pic]20
With: x1[pic]0 , x2[pic]0 , and x3[pic]0
|Show your work here. |
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|Write the augmented matrix for the maximization problem: |
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|[pic] |
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|Find the transpose of that matrix: |
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|[pic] |
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|Write the minimization problem from the transposed matrix: |
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|Minimize z = 18y1 + 20y2 |
|Subject to: 7y1 + 4y2 ≥ 8 |
|6y1 + 5y2 ≥ 3 |
|8y1 + 10y2 ≥ 1 |
|With: y1 ≥ 0, y2 ≥ 0 |
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10. Use the simplex method to solve.
Find: y1[pic]0, y2[pic]0 such that
3y1+y2[pic]12
y1+4y2[pic]16
and w=2y1+y2 is minimized.
|Show your work here. |
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|Form the augmented matrix: |
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|[pic] |
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|Form the transpose of that matrix: |
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|[pic] |
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|Write the corresponding maximization problem: |
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|Maximize w = 12x1 + 16x2 |
|Subject to: 3x1 + x2 ≤ 2 |
|x1 + 4x2 ≤ 1 |
|x1 ≥ 0, x2 ≥ 0 |
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|The augmented matrix for the maximization problem is: |
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|[pic] |
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|[pic] |
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|The solutions come from the columns for the slack variables: |
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|W = 100/11, with y1 = 32/11, y2 = 36/11 |
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