Matrix Algebra Tutor - Worksheet 4 â Row Equivalent
Algebra 2 – Matrix Algebra Tutor Worksheet 4 – Row Equivalent Matrices
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Algebra 2 – Matrix Algebra Tutor - Worksheet 4 – Row Equivalent Matrices
1. Create a row equivalent matrix with the manipulations shown below.
2. Create a row equivalent matrix with the manipulations shown below.
3. Create a row equivalent matrix with the manipulations shown below.
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4. Create a row equivalent matrix with the manipulations shown below.
5. Create a row equivalent matrix with the manipulations shown below.
6. Create a row equivalent matrix with the manipulations shown below.
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7. Create a row equivalent matrix with the manipulations shown below.
8. Create a row equivalent matrix with the manipulations shown below.
9. Create a row equivalent matrix with the manipulations shown below.
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10. Create a row equivalent matrix with the manipulations shown below.
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Answers - Algebra 2 – Matrix Algebra Tutor - Worksheet 4 – Row Equivalent Matrices
1. Create a row equivalent matrix with the manipulations shown below.
The symbol 𝑅1 ↔ 𝑅3 means interchange Row 1 and Row 3. The symbol ← 3𝑅2 means multiply Row 2 by 3 and put the results in Row 2. The math of this manipulation looks like this: 2 4 [3(6) 3(0) 3 −1
1 ⋮ 18 3(2) ⋮ 3(−11)] 4 ⋮ 17
The row equivalent matrix looks like this: Answer: 2 4 [18 0 3 −1
1 ⋮ 18 6 ⋮ −33] 4 ⋮ 17
2. Create a row equivalent matrix with the manipulations shown below.
1
1
The symbol ← 𝑅1 means multiply Row 1 by , and put the result in Row 1. The 3
1
3
1
symbol ← 𝑅2 means multiply Row 2 by , and put the result in Row 2. And the 2
2
1
1
7
7
symbol ← − 𝑅3 means multiply Row 3 by − , and put the result in Row 3. The math of these manipulations looks like this: 6 © MathTutorDVD.com
1 1 1 1 (6) (3) (−3) ⋮ (27) 3 3 3 3 1 1 1 1 (2) (−4) (−2) ⋮ (12) 2 2 2 2 1 1 1 1 − (7) − (21) − (−14) ⋮ − (28) [ 7 ] 7 7 7
The row equivalent matrix looks like this: Answer: 2 1 [ 1 −2 −1 −3
−1 ⋮ 9 −1 ⋮ 6 ] 2 ⋮ −4
3. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 3𝑅2 − 𝑅1 means multiply Row 2 by 3, subtract Row 1, and put the result in Row 1. The symbol ← −2𝑅2 + 𝑅3 means multiply Row 2 by −2, add Row 3, and put the results in Row 3. The math of these manipulations looks like this: 3(−2) − (−1) ⋮ 3(13) − 3 ] −2 ⋮ 13 −2(−2) + 0 ⋮ −2(13) + 5
3(−3) − 1 3(4) − 0 −3 4 [ −2(−3) + (−2) −2(4) + 2
The row equivalent matrix looks like this: Answer: −10 12 [ −3 4 4 −6
−5 ⋮ 36 −2 ⋮ 13 ] 4 ⋮ −21 7
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4. Create a row equivalent matrix with the manipulations shown below.
The symbol ← −2𝑅1 means multiply Row 1 by −2, and put the result in Row 1. The symbol 𝑅2 ↔ 𝑅3 means interchange Row 2 and Row 3. The math of these manipulations looks like this: −2(3) −2(2) −2(1) ⋮ −2(18) [ 4 ] 6 5 ⋮ 15 2 4 3 ⋮ 20 The row equivalent matrix looks like this: Answer: −6 −4 [4 6 2 4
−2 ⋮ −36 5 ⋮ 15 ] 3 ⋮ 20
5. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 + 𝑅2 means add Row 1 and Row2, and put the result in Row 2. 1
1
2
2
The symbol ← 𝑅3 means multiply Row 3 by , and put the result in Row 3. The math of these manipulations looks like this:
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−5 2 5 + (−5) 2 + (−3) [ 1 1 (−4) (4) 2 2
−1 ⋮ −13 −1 + 3 ⋮ −13 + 12 ] 1 1 (−6) ⋮ (14) 2 2
The row equivalent matrix looks like this: Answer: −5 2 −1 [ 0 −1 2 −2 2 −3
⋮ −13 ⋮ −1 ] ⋮ 7
6. Create a row equivalent matrix with the manipulations shown below.
1
1
2 1
2 1
3 1
3 1
4
4
The symbol ← − 𝑅1 means multiply Row 1 by − , and put the result in Row 1. The symbol ← − 𝑅2 means multiply Row 2 by − , and put the result in Row 2. The symbol ← − 𝑅3 means multiply Row 3 by − , and put the result in Row 3. The math of these manipulations looks like this: 1 1 1 1 − (6) − (4) − (8) ⋮ − (−32) 2 2 2 2 1 1 1 1 − (9) − (−6) − (−3) ⋮ − (15) 3 3 3 3 1 1 1 1 − (−8) − (−4) − (12) ⋮ − (−24) [ 4 ] 4 4 4 The row equivalent matrix looks like this: Answer:
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−3 [−3 2
−2 2 1
−4 ⋮ 16 1 ⋮ −5] −3 ⋮ 6
7. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 + 𝑅2 means add Row 1 and Row 2, and put the result in Row 2. The symbol ← 𝑅2 + 𝑅3 means add Row 2 and Row 3, and put the result in Row 3. The math of these manipulations looks like this: −1 3 7 ⋮ −9 3+2 7 + 8 ⋮ −9 + 18] [ −1 + 1 1 + (−1) 2 + (−3) 8 + 9 ⋮ 18 + 21 The row equivalent matrix looks like this: Answer: −1 3 7 ⋮ [0 5 15 ⋮ 0 −1 17 ⋮
−9 9] 39
8. Create a row equivalent matrix with the manipulations shown below.
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The symbol ← −𝑅1 means multiply Row 1 by −1, and put the result in Row 1. The symbol ← −2𝑅3 + 𝑅2 means multiply Row 3 by −2, add Row 2, and put the result in Row 2. The math of these manipulations looks like this: −(−3) [−2(1) + 2 1
−(1) −2(5) + 7 5
−(−2) ⋮ −(14) −2(−6) + (−5) ⋮ −2(12) + (−17)] −6 ⋮ 12
The row equivalent matrix looks like this: Answer: 3 −1 2 ⋮ −14 [0 −3 7 ⋮ −41 ] 1 5 −6 ⋮ 12
9. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 − 2𝑅2 means multiply Row 2 by −2,add Row 1, and put the result in Row 2. The symbol ← 𝑅3 + 2𝑅2 means multiply Row 2 by 2, add Row 3, and put the result in Row 3. The math of these manipulations looks like this:
6 −2 5 ⋮ −11 [ 6 − 2(3) −2 − 2(8) 5 − 2(−7) ⋮ −11 − 2(2)] −9 + 2(3) −6 + 2(8) 4 + 2(−7) ⋮ −16 + 2(2) The row equivalent matrix looks like this: Answer: 6 −2 [ 0 −18 −3 10
5 19 −10 11
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⋮ −11 ⋮ −15] ⋮ −12
10. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 + 𝑅2 means add Row 1 and Row 2, putting the result in Row 2. The symbol ← 2𝑅1 + 𝑅3 means multiply Row 1 by 2, add Row 3, and put the result in Row 3. The math of these manipulations looks like this:
−5 7 2 ⋮ 7 2 + (−6) ⋮ 7 + 4 7+8 [ −5 + 5 ] 2(−5) + 10 2(7) + (−14) 2(2) + 5 ⋮ 2(7) + 11 The row equivalent matrix looks like this: Answer: −5 7 2 ⋮ [ 0 15 −4 ⋮ 0 0 9 ⋮
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Algebra 2 – Matrix Algebra Tutor - Worksheet 4 – Row Equivalent Matrices
1. Create a row equivalent matrix with the manipulations shown below.
2. Create a row equivalent matrix with the manipulations shown below.
3. Create a row equivalent matrix with the manipulations shown below.
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4. Create a row equivalent matrix with the manipulations shown below.
5. Create a row equivalent matrix with the manipulations shown below.
6. Create a row equivalent matrix with the manipulations shown below.
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7. Create a row equivalent matrix with the manipulations shown below.
8. Create a row equivalent matrix with the manipulations shown below.
9. Create a row equivalent matrix with the manipulations shown below.
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10. Create a row equivalent matrix with the manipulations shown below.
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Answers - Algebra 2 – Matrix Algebra Tutor - Worksheet 4 – Row Equivalent Matrices
1. Create a row equivalent matrix with the manipulations shown below.
The symbol 𝑅1 ↔ 𝑅3 means interchange Row 1 and Row 3. The symbol ← 3𝑅2 means multiply Row 2 by 3 and put the results in Row 2. The math of this manipulation looks like this: 2 4 [3(6) 3(0) 3 −1
1 ⋮ 18 3(2) ⋮ 3(−11)] 4 ⋮ 17
The row equivalent matrix looks like this: Answer: 2 4 [18 0 3 −1
1 ⋮ 18 6 ⋮ −33] 4 ⋮ 17
2. Create a row equivalent matrix with the manipulations shown below.
1
1
The symbol ← 𝑅1 means multiply Row 1 by , and put the result in Row 1. The 3
1
3
1
symbol ← 𝑅2 means multiply Row 2 by , and put the result in Row 2. And the 2
2
1
1
7
7
symbol ← − 𝑅3 means multiply Row 3 by − , and put the result in Row 3. The math of these manipulations looks like this: 6 © MathTutorDVD.com
1 1 1 1 (6) (3) (−3) ⋮ (27) 3 3 3 3 1 1 1 1 (2) (−4) (−2) ⋮ (12) 2 2 2 2 1 1 1 1 − (7) − (21) − (−14) ⋮ − (28) [ 7 ] 7 7 7
The row equivalent matrix looks like this: Answer: 2 1 [ 1 −2 −1 −3
−1 ⋮ 9 −1 ⋮ 6 ] 2 ⋮ −4
3. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 3𝑅2 − 𝑅1 means multiply Row 2 by 3, subtract Row 1, and put the result in Row 1. The symbol ← −2𝑅2 + 𝑅3 means multiply Row 2 by −2, add Row 3, and put the results in Row 3. The math of these manipulations looks like this: 3(−2) − (−1) ⋮ 3(13) − 3 ] −2 ⋮ 13 −2(−2) + 0 ⋮ −2(13) + 5
3(−3) − 1 3(4) − 0 −3 4 [ −2(−3) + (−2) −2(4) + 2
The row equivalent matrix looks like this: Answer: −10 12 [ −3 4 4 −6
−5 ⋮ 36 −2 ⋮ 13 ] 4 ⋮ −21 7
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4. Create a row equivalent matrix with the manipulations shown below.
The symbol ← −2𝑅1 means multiply Row 1 by −2, and put the result in Row 1. The symbol 𝑅2 ↔ 𝑅3 means interchange Row 2 and Row 3. The math of these manipulations looks like this: −2(3) −2(2) −2(1) ⋮ −2(18) [ 4 ] 6 5 ⋮ 15 2 4 3 ⋮ 20 The row equivalent matrix looks like this: Answer: −6 −4 [4 6 2 4
−2 ⋮ −36 5 ⋮ 15 ] 3 ⋮ 20
5. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 + 𝑅2 means add Row 1 and Row2, and put the result in Row 2. 1
1
2
2
The symbol ← 𝑅3 means multiply Row 3 by , and put the result in Row 3. The math of these manipulations looks like this:
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−5 2 5 + (−5) 2 + (−3) [ 1 1 (−4) (4) 2 2
−1 ⋮ −13 −1 + 3 ⋮ −13 + 12 ] 1 1 (−6) ⋮ (14) 2 2
The row equivalent matrix looks like this: Answer: −5 2 −1 [ 0 −1 2 −2 2 −3
⋮ −13 ⋮ −1 ] ⋮ 7
6. Create a row equivalent matrix with the manipulations shown below.
1
1
2 1
2 1
3 1
3 1
4
4
The symbol ← − 𝑅1 means multiply Row 1 by − , and put the result in Row 1. The symbol ← − 𝑅2 means multiply Row 2 by − , and put the result in Row 2. The symbol ← − 𝑅3 means multiply Row 3 by − , and put the result in Row 3. The math of these manipulations looks like this: 1 1 1 1 − (6) − (4) − (8) ⋮ − (−32) 2 2 2 2 1 1 1 1 − (9) − (−6) − (−3) ⋮ − (15) 3 3 3 3 1 1 1 1 − (−8) − (−4) − (12) ⋮ − (−24) [ 4 ] 4 4 4 The row equivalent matrix looks like this: Answer:
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−3 [−3 2
−2 2 1
−4 ⋮ 16 1 ⋮ −5] −3 ⋮ 6
7. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 + 𝑅2 means add Row 1 and Row 2, and put the result in Row 2. The symbol ← 𝑅2 + 𝑅3 means add Row 2 and Row 3, and put the result in Row 3. The math of these manipulations looks like this: −1 3 7 ⋮ −9 3+2 7 + 8 ⋮ −9 + 18] [ −1 + 1 1 + (−1) 2 + (−3) 8 + 9 ⋮ 18 + 21 The row equivalent matrix looks like this: Answer: −1 3 7 ⋮ [0 5 15 ⋮ 0 −1 17 ⋮
−9 9] 39
8. Create a row equivalent matrix with the manipulations shown below.
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The symbol ← −𝑅1 means multiply Row 1 by −1, and put the result in Row 1. The symbol ← −2𝑅3 + 𝑅2 means multiply Row 3 by −2, add Row 2, and put the result in Row 2. The math of these manipulations looks like this: −(−3) [−2(1) + 2 1
−(1) −2(5) + 7 5
−(−2) ⋮ −(14) −2(−6) + (−5) ⋮ −2(12) + (−17)] −6 ⋮ 12
The row equivalent matrix looks like this: Answer: 3 −1 2 ⋮ −14 [0 −3 7 ⋮ −41 ] 1 5 −6 ⋮ 12
9. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 − 2𝑅2 means multiply Row 2 by −2,add Row 1, and put the result in Row 2. The symbol ← 𝑅3 + 2𝑅2 means multiply Row 2 by 2, add Row 3, and put the result in Row 3. The math of these manipulations looks like this:
6 −2 5 ⋮ −11 [ 6 − 2(3) −2 − 2(8) 5 − 2(−7) ⋮ −11 − 2(2)] −9 + 2(3) −6 + 2(8) 4 + 2(−7) ⋮ −16 + 2(2) The row equivalent matrix looks like this: Answer: 6 −2 [ 0 −18 −3 10
5 19 −10 11
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⋮ −11 ⋮ −15] ⋮ −12
10. Create a row equivalent matrix with the manipulations shown below.
The symbol ← 𝑅1 + 𝑅2 means add Row 1 and Row 2, putting the result in Row 2. The symbol ← 2𝑅1 + 𝑅3 means multiply Row 1 by 2, add Row 3, and put the result in Row 3. The math of these manipulations looks like this:
−5 7 2 ⋮ 7 2 + (−6) ⋮ 7 + 4 7+8 [ −5 + 5 ] 2(−5) + 10 2(7) + (−14) 2(2) + 5 ⋮ 2(7) + 11 The row equivalent matrix looks like this: Answer: −5 7 2 ⋮ [ 0 15 −4 ⋮ 0 0 9 ⋮
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