Systems of Linear Equations
Systems of Linear
EquationsSystems of Linear
EquationsBy: Tri Atmojo Kusmayadi and Mardiyana
Standard of Competency:
Understanding the properties of
systems of linear equations,
matrices, and their use in
problem solving
3
2 Definition
We define a linear equation in the n variables x , x , …, x to be one that can be expressed in
1 2 n
the form
a x + a x + … + a x = b
1
1
2 2 n n where a , a , …, a and b are real constants.
1 2 n
The variables in a linear equation are sometimes called the unknowns.
Definition
A solution of a linear equation a x + a x + … +
1
1
2
2
a x = b is a sequence of n numbers s , s , …, s
n n 1 2 n
such that the equation is satisfied when we substitute x = s , x = s , …, x = s .
1
1
2 2 n n
The set of all solutions of the equation is called its
solution set or the general solution of the
equation Examples :
Find the solution set of (a) and (b)
Solutions : (a) (b) Linear equations Systems
A finite set of linear equations in the variables x , x , …, x
1 2 n
is called a system of linear equations or a linear systems .
A sequence of n numbers s , s , …, s is called a solution
1 2 n
of the linear system if x = s , x = s , …, x = s is a
1
1
2 2 n n solution of every equation in the linear system.
For example, the system
has the solution
However, is not the solution
Every system of linear The lines l and may be 1 l 2 equations has no parallel, in which case there is no solutions, or has exactly solution to the system. intersection and consequently no one solution, or has infinitely many solutions.
The lines l and may intersect solution. the system has exactly one at only one point, in which case 1 l 2 The lines l and may are infinitely many points of coincide, in which case there 1 l 2 The General Form of Linear Systems
An arbitrary system of m linear equations in n variables can be written as
a x + a x + … + a x = b
11
1
12 2 1n n
1 a x + a x + … + a x = b
21
1
22 2 2n n
2 : : : : a x + a x + … + a x = b m1 1 m2 2 mn n m
where x , x , …, x are the unknowns and the subscripted
1 2 n a’s and b’s denote real constants. Augmented Matrices
A system of m linear equations in n unknowns can be written as follows:
⎥ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎢ ⎣ ⎡ m mn m m n n b a a a b a a a b a a a
2
1
2
2
22
21
1
1
12
11
L M M O M M L L
For example, the augmented matrix for the system of equations Elementary Row Operations
Three types of operations to eliminate unknowns: 1.
Multiply an equation through by a nonzero constant.
2. Interchange two equations.
3. Add a multiple of one equation to another.
Three types of operations on the rows of the augmented matrix:
1. Multiply a row through by a nonzero constant.
2. Interchange two rows. Example
In the following column below we solve a system of linear equations by
1
1
2
9
2
4
3
3
operating on the equations in the systems x + y + 2z = 9 2x + 4y – 3z = 1
6
5
− −
⎢ ⎢ ⎣ ⎡
operating on the rows of the augmented matrix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
In the following column below we solve a system of linear equations by
3x + 6y – 5z = 0
1 Add -2 times the first row to the second row and add -3 times the first row to the third row to obtain
⎥ ⎥ ⎥ ⎦ ⎤ ⎢
⎢ ⎢ ⎣ ⎡
− − − −
27
11
3
17
7
2
9
2
1
1 Add -2 times the first
equation to the second equation and add -3 times the first equation to the third equation to obtain
x + y + 2z = 9 2y – 7z = -17 3y – 11z = -27
Multiply the second row by ½ to obtain ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
3y – 11z = -27 Add -3 times the second equation to the third to obtain x + y + 2z = 9 y – 7/2 z = - 17/2
1
2
9
1
− − − − 2 3 2 1 2 17 2 7
⎢ ⎢ ⎣ ⎡
Add -3 times the second row to the third to obtain ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
1 2 17 2 7 Multiply the second equation by ½ to obtain x + y + 2z = 9 y – 7/2 z = - 17/2
⎢ ⎢ ⎣ ⎡
1
2
9
1
3
11
27
− − − −
1 Multiply the third row by-2 to obtain ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
⎢ ⎢ ⎣ ⎡
Add -1 times the second equation to the first to obtain x + 11/2 z = 35/2 y – 7/2 z = - 17/2
1
1
3
− −
⎢ ⎢ ⎢ ⎡
Add -1 times the second row to the first to obtain ⎥ ⎥ ⎥ ⎤
1 2 17 2 7 Multiply the third equation by -2 to obtain x + y + 2z = 9 y – 7/2 z = - 17/2 z = 3
− −
1
2
9
1
1
3
1 2 17 2 7 2 35 2 11 Add -11/2 time the third row to the first and 7/2 time the third row to the second to obtain
Add -11/2 time the third equation to the first and 7/2 time the third equation to the second to obtain
x = 1 y = 2 z = 3
⎥ ⎥ ⎥ ⎦ ⎤ ⎢
⎢ ⎢ ⎣ ⎡
3
1
2
1
1
1 Thus the solution of the system of linear equations is Gaussian Elimination Gaussian Elimination
Reduced Row-Echelon FormA matrix is said to be in reduced row-echelon form , if the following
properties are satisfied:1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. (we call this a leading 1 ).
2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix .
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.
4. Each column that consists a leading 1 has zeros everywhere else .
A matrix having properties 1, 2 and 3 is said to be in row-echelon
form . (Thus, a matrix in reduced row-echelon form is of necessity in
row-echelon form, but not conversely.)Homogeneous Linear Systems
A system of linear equations is said to be homogeneous if the constant terms are all zero, the system has the form
a x + a x + … + a x = 0
11
1
12 2 1n n a x + a x + … + a x = 0
21
1
22 2 2n n : : : : a x + a x + … + a x = 0 m1 1 m2 2 mn n
Every homogeneous system of linear equations is consistent, since all such systems have x = 0, x = 0, …,
1
2 x = 0 as a solution. This solution is called the trivial n
Example
5 = 0 x
5 = 0
4 + x
3 + x
5 = 0 x
3 - x
2 - 2x
1 + x
Solve the following homogeneous system of linear equations by Gauss-Jordan eliminations.
2x
3 – 3x
2 + 2x
1 - x
5 = 0 -x
3 + x
2 - x
1 + 2x
4 + x
Solution: The augmented matrix for the system is
1
1
1
1
1
1
⎢ ⎡
⎥ ⎥ ⎤ ⎢
2
1
1
⎥ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎢ ⎣ ⎡
2
3
1
1
1
2
1
1
1
1
− −
− − − −1 The corresponding system of equations is
x + x + x = 0
1
2
5 x + x = 0
3
5 x = 0
x = -s – t, x = s, x = -t, x = 0, x = t,
1
2
3
4
5 where s and t are parameters.
Theorem
A homogeneous system of linear equations with
more unknowns than equations has infinitely many
solutions. Problem 1
Solve the following system of nonlinear equations for x, y and z.
2
2
2 x + y + z = 6
2
2
2 x - y + 2z = 2
2
2
2 2x + y - z = 3
Problem 2
Show that the following nonlinear system has eighteen ≤ α ≤ 2π, 0 ≤ β ≤ 2π , 0 ≤ γ ≤ 2 π. solutions if 0 sin α + 2 cos β + 3 tan γ = 0
Matrices and matrix
OperationsMatrices and matrix
OperationsBy: Tri Atmojo K and Mardiyana Mathematics Education Sebelas Maret University In the previous Section we used rectangular arrays of numbers, called augmented matrices, to abbreviate systems of linear equations.
However, rectangular arrays of numbers occur in other contexts as well. For example,
the following rectangular array with three rows and seven columns might describe the
number of hours that a student spent studying three subjects during a certain week:If we suppress the headings, then we are left with the following
rectangular array of numbers with three rows and seven columns,
called a “matrix”:Definition
A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix.
Definition
Two matrices are defined to be equal if the have the same size and their corresponding entries are equal.
Definition
If A and B are matrices of the same size, then the sum A + B
Definition
If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of A by c.
Definition
If A is an m x r matrix and B is an r x n matrix, then the
product AB is the m x n matrix whose entries are
determined as follows. To find the entry in row i and column j of AB, single out row I from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together and then add up the resulting products.
For example, if A, B, and C are the matrices in the earlier Example, then
Definition
I f A is any m x n m at rix, t hen t he t r a n spose of A,
T
denot ed by A , is defined t o be t he n x m m at rix t hat result s from int erchanging t he row s and
T
colum ns of A, t hat is, t he first colum n of A is t he
T
first row of A, t he second colum n of A is t he second row of A, and so fort h.
Definition
I f A is a square m at rix, t hen t he t r a ce of A, denot ed by t r( A) , is defined t o be t he sum of t he
Definition
I f A is a square m at rix, and if a m at rix B of t he sam e size can be found such t hat AB = BA = I , w here I is an ident it y m at rix, t hen A is said t o be in ve r t ible and B is called an in ve r se of A.
Theorem
I f B and C are bot h inverses of t he m at rix A, t hen B = C.
Theorem
Theorem
If A is an invertible matrix, then
- 1
- 1 -1
a). A is invertible and (A ) = A
n n -1 -1 n
b). A is invertible and (A ) = (A ) for n = 1,2, …
c). For any nonzero scalar k, the matrix kA is invertible and
- 1 -1 (kA) = (1/k) A .
T T -1 -1 T d). A is invertible and (A ) = (A ) .
Proof: Exercise for student.
Problem 1 T
A square matrix A is called symmetric if A = A and skew-
T
symmetric if A = -A. Show that if B is a square matrix, then
T T a). BB and B + B are symmetric.
T b). B – B is skew-symmetric.
Problem 2 Let A be a square matrix.
- 1
2
3
4
5 a). Show that (I – A) = I + A + A + A + A if A = 0.
Elementary Matrices and a Method for Finding A -1 Definition
An n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix I
n
by performing a single elementary row operation.
Theorem
If the elementary matrix A results from performing a certain row operation on I
m
and if A an m x n matrix, then the product EA is the matrix that results when this same row
Theorem
Every elementary matrix is invertible, and the inverse is also an elementary matrix.
Theorem
If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false.
a). A is invertible.
b). Ax = 0 has only the trivial solution.
c). The reduced row-echelon form of A is I .
n d). A is expressible as a product of elementary matrices.
Remark:
To find the inverse of an invertible matrix A,
we must find a sequence of elementary row
operations that reduces A to the identity and
then perform this same sequence of
operations on I n to obtain A- 1 .
Example : Find the inverse of
⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎣ ⎡ =
8
1
3
5
2
3
2