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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation
The variables and equality signs do not take part.
For a m × n system a1 1 x 1 + a1 2 x 2 + · · · + a1 n x n a2 1 x1 + a2 2 x2 + · · · + a2 n xn = b1 = b2 .
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. am 1 x1 + am 2 x2 + · · · + am n xn = bm the augmented matrix is a rectangular array a1 1 a2 1
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.
. am 1 a1 2 ··· a1 n a2 2 ··· a2 n am 2 ··· am n .
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. .
.
. .
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. b1 b2 .
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. bm The vertical line  separate coeﬃcients & constant terms Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix  example Example. Write down the augmented matrix of x1 + 2x2 + x3 3x1 − x2 − 3x3 2x1 + 3x2 + x3 =3 = −1
=4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix  example Example. Write down the augmented matrix of x1 + 2x2 + x3 3x1 − x2 − 3x3 2x1 + 3x2 + x3 Solution. The augmented matrix is 1
1 2 3 −1 −3 23
1 =3 = −1
=4 3 −1 4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Motivation 2x − y = 9 Example. Solve −x + 7y = −10 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Motivation = 2x − y 9 Example. Solve −x + 7y = −10 2x − y
Solution. − x + 7y = 9 = −10 − − − − − −−→
− − − − −− = −10 = 9 −2x + 14y 2x − y − x + 7y 2x − y = −20 = 9 − − − − − −−→
− − − − −− −2x + 14y 13y = −20 = −11 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Motivation = 2x − y 9 Example. Solve −x + 7y = −10 2x − y
Solution. − x + 7y = 9 = −10 − − − − − −−→
− − − − −− −2x + 14y 2x − y = −20 = 9 − − − − − −−→
− − − − −− In terms of augmented matrix,
2 −1 = −10 = 9 − x + 7y 2x − y −1 9 7 −10 −2x + 14y 13y = −20 = −11 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Motivation = 2x − y 9 Example. Solve −x + 7y = −10 Interchange the equations 2x − y
Solution. − x + 7y = 9 = −10 − − − − − −−→
− − − − −− − x + 7y 2x − y −2x + 14y 2x − y = −20 = 9 − − − − − −−→
− − − − −− −2x + 14y 13y = 9 −1
7 −1 −→ 2
−10
9 = −20 = −11 Interchange rows In terms of augmented matrix,
2 −1 −10 = 7
−1 −10 9 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Motivation = 2x − y 9 Example. Solve −x + 7y = −10 Multiply 2 to 1st equation 2x − y
Solution. − x + 7y = 9 = −10 − − − − − −−→
− − − − −− −2x + 14y 2x − y = −20 = 9 − − − − − −−→
− − − − −− −10 = 9 −1
7
14
−1 −1 −→ 2
−10 −20 9
9 −2x + 14y 13y = −20 = −11 Multiply 2 to 1st row In terms of augmented matrix,
2 −1 −2 2 = − x + 7y 2x − y 7
−1 −10 −→
9 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Motivation = 2x − y 9 Example. Solve −x + 7y = −10 Add 1st equation to the 2nd 2x − y
Solution. − x + 7y = 9 = −10 − − − − − −−→
− − − − −− −2x + 14y 2x − y = −20 = 9 − − − − − −−→
− − − − −− −10 = 9 −1
7
14
−1 −1 −→ 2
−10 −20 −2 −→ 9
0
9 −2x + 14y 13y = −20 = −11 Add 1st row to the 2nd In terms of augmented matrix,
2 −1 −2 2 = − x + 7y 2x − y 7
−1
14
13 −10 −→
9 −20 −11 Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Elementary Row Operations The above operations are suﬃcient to solve any system of linear
equations! Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Elementary Row Operations The above operations are suﬃcient to solve any system of linear
equations!
Elementary Row Operations
I. Interchange two rows
II. Multiply a row by a nonzero real number
III. Replace a row by its sum with a multiple of another row Notation
I. Ri ↔ Rj
II. αRi
III. αRi + Rj (interchange ith & jth rows)
(Multiply ith row by α)
(Replace jth row by αRi + Rj ) or
(Add αRi to Rj ) Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Methodology Locate the 1st column containing a nonzero entry
Pick a row with a nonzero entry to be pivotal row
Perform row operations to eliminate other entries in the column
Repeat the steps to the remaining smaller matrix. Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Methodology Locate the 1st column containing a nonzero entry
Pick a row with a nonzero entry to be pivotal row
Perform row operations to eliminate other entries in the column
Repeat the steps to the remaining smaller matrix. 1 3 2 2 1 −1 −3 3 1 3 −1 4 Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Methodology Locate the 1st column containing a nonzero entry
Pick a row with a nonzero entry to be pivotal row
Perform row operations to eliminate other entries in the column
Repeat the steps to the remaining smaller matrix. 1 3 2 2 1 −1 −3 3 1 3 −1 4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry
Pick a row with a nonzero entry to be pivotal row
Perform row operations to eliminate other entries in the column
Repeat the steps to the remaining smaller matrix. 1 3 2 2 1 −1 −3 3 1 3
1 −1 −→ 0 4
0 2 1 −7 −6 −1 −1 3 −10 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry
Pick a row with a nonzero entry to be pivotal row
Perform row operations to eliminate other entries in the column
Repeat the steps to the remaining smaller matrix. 1 3 2 2 1 −1 −3 3 1 3
1 −1 −→ 0 4
0 2 1 −7 −6 −1 −1 3 −10 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry
Pick a row with a nonzero entry to be pivotal row
Perform row operations to eliminate other entries in the column
Repeat the steps to the remaining smaller matrix. 1 3 2 2 1 −1 −3 3 1 3
1
1 2 −1 −→ 0 −7 −6 4
0 −1 − 1 1
1 2 0 − 7 −6
−→ 1
0 0 −7 3 −10
−2 3 −10 4
−7 Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 0 −1 −1 1 0 1 1 1 1 6 2 4 1 −2 −1 3 1 −2 2 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 0 −1 −1 1 0 1 1 1 1 6 2 4 1 −2 −1 3 1 −2 2 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 2 4 1 −2 −1 3 1 −2 2 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 2 −1 −4 −13 0 −2 −5 −1 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 2 −1 −4 −13 0 −2 −5 −1 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 2 −1 4 −13 0 −2 −5 −1 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 0 −3 −2 −13 0 0 −3 −3 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 0 −3 −2 −13 0 0 −3 −3 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 0 −3 −2 −13 0 0 −3 −3 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 0 −3 −2 −13 0 0 0 −1 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations  Example Example. Solve −x2 − x3 + x4 = 0 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = 6 = −1 = 3 Solution. The augmented matrix is 1 1 1 1 6 0 −1 −1 1 0 0 0 −3 −2 −13 0 0 0 −1 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form  Example 1 Augmented matrix in Triangular form  tell solution immediately.
Example. Solve the system −x2 − x3 + x4
x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 =
=
=
= 0
6
−1
3 if its augmented matrix is reduced to 1 0 0 0 1
1
0
0 1
1
1
0 1
−1
2
3 1 6 0
13 3
2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form  Example 1 Augmented matrix in Triangular form  tell solution immediately.
Example. Solve the system −x2 − x3 + x4
x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 =
=
=
= 0
6
−1
3 if its augmented matrix is reduced to 1 0 0 0 1
1
0
0 1
1
1
0 The solution set is {(−4, −1, 3, 2)}. 1
−1
2
3 1 6 0
13 3
2 Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Lead & Free variables There are two types of variables in the 2nd example.
Lead variables & Free variables
A variable corresponding to a leading one is called a
lead variable
All remaining variables are called free variables Remark
A system has inﬁnitely many solutions if and only if it has free variable(s). Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Type of Solutions Observations
(0 0 ··· 0 1) means an inconsistent system Number of lead variables = number of variables (i.e. no free
variable)
Each variable is uniquely determined. ∴ Exactly one solution.
Free variable ⇒ inﬁnitely many solutions
Remarks.
1. The number of lead variables = the number of leading one’s.
(Recall what is a lead variable.)
2. In Section 1.2.4 of Nicholson, the number of leading one’s is called the "rank of a
matrix". We do not use this name in this chapter because we shall deﬁne "rank" in
another way. These two deﬁnitions are equivalent (i.e. deﬁning the same object). Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Math1111 Type of Solutions Theorem
For a system of linear equations there are exactly three possibilities:
No solution.
A unique solution.
Inﬁnitely many solution Remark. The solution type of a linear system can be determined from
the row echelon form of its augmented matrix. (How? See the last
slide.) Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Question
Is row echelon form unique? Math1111 Reduced Row Echelon Form Chapter 1. Matrices and Systems of Equations Systems of Linear Equations Question
Is row echelon form unique?
Ans. NO Math1111 Reduced Row Echelon Form ...
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