Systems of Linear Equations

Systems of Linear Equations

By:Tri Atmojo Kusmayadi and Mardiyana

Mathematics Education Sebelas Maret University

Standard of Competency:

Understanding the properties of systems of linear equations,

matrices, and their use in problem solving

Basic Competencies:31 2

A. Introduction to Systems of Linear Equations

DefinitionWe define a linear equation in the n variables x1, x2, …, xn to be one that can be expressed in the form

a1x1 + a2x2 + … + anxn = bwhere a1, a2, …, an and b are real constants. The variables in a linear equation are sometimes called the unknowns.

Examples :

DefinitionA solution of a linear equation a1x1 + a2x2 + … + anxn = b is a sequence of n numbers s1, s2, …, snsuch that the equation is satisfied when we substitute x1 = s1, x2 = s2, …, xn = sn.

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 x1, x2, …, xnis called a system of linear equations or a linear systems.

A sequence of n numbers s1, s2, …, sn is called a solution of the linear system if x1 = s1, x2 = s2, …, xn = sn is a solution of every equation in the linear system.

For example, the system

has the solution

However,is not the solution

Every system of linear equations has no solutions, or has exactly one solution, or has infinitely many solutions.

The lines l1 and l2 may be parallel, in which case there is no intersection and consequently no solution to the system.

The lines l1 and l2 may intersect at only one point, in which case the system has exactly one solution.

The lines l1 and l2 may coincide, in which case there are infinitely many points of intersection and consequently infinitely many solutions to the system.

The General Form of Linear SystemsAn arbitrary system of m linear equations in n variables can be written as

a11x1 + a12x2 + … + a1nxn = b1a21x1 + a22x2 + … + a2nxn = b2: : : :am1x1 + am2x2 + … + amnxn = bm

where x1, x2, …, xn are the unknowns and the subscripted a’s and b’s denote real constants.

Augmented Matrices

A system of m linear equations in n unknowns can be written as follows:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

mmnmm

n

n

baaa

baaabaaa

L

MMOMM

L

L

21

222221

111211

This is called the augmented matrix for the system.

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.3. Add a multiple of one row to another row.

Example

In the following column below we solve a system of linear equations by operating on the equations in the systems

x + y + 2z = 92x + 4y – 3z = 13x + 6y – 5z = 0

In the following column below we solve a system of linear equations by operating on the rows of the augmented matrix

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

056313429211

Add -2 times the first row to the second row and add -3 times the first row to the third row to obtain

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

271130177209211

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 = 92y – 7z = -173y – 11z = -27

Multiply the second row by ½to obtain

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

27113010

9211

217

27

Multiply the second equation by ½to obtain

x + y + 2z = 9y – 7/2 z = - 17/2

3y – 11z = -27

Add -3 times the second equation to the third to obtain

x + y + 2z = 9y – 7/2 z = - 17/2-- ½ z = -3/2

Add -3 times the second row to the third to obtain

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

23

21

217

27

0010

9211

Multiply the third row by-2 to obtain

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−3100

109211

217

27

Multiply the third equation by -2 to obtain

x + y + 2z = 9y – 7/2 z = - 17/2

z = 3

Add -1 times the second equation to the first to obtain

x + 11/2 z = 35/2y – 7/2 z = - 17/2

z = 3

Add -1 times the second row to the first to obtain

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−3100

1001

217

27

235

211

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 = 1y = 2

z = 3 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310020101001

Thus the solution of the system of linear equations is x = 1, y = 2, z = 3.

Gaussian Elimination

Gaussian Elimination

Reduced Row-Echelon Form

A 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

a11x1 + a12x2 + … + a1nxn = 0a21x1 + a22x2 + … + a2nxn = 0

: : : :am1x1 + am2x2+ … + amnxn = 0

Every homogeneous system of linear equations is consistent, since all such systems have x1= 0, x2 = 0, …, xn = 0 as a solution. This solution is called the trivial solution, if there are other solutions, they are called nontrivial solution.

Example

Solve the following homogeneous system of linear equations by Gauss-Jordan eliminations.

2x1 + 2x2 - x3 + x5 = 0-x1 - x2 + 2x3 – 3x4 + x5 = 0x1 + x2 - 2x3 - x5 = 0

x3 + x4 + x5 = 0

Solution: The augmented matrix for the system is

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−

−

011100010211013211010122

Reducing this matrix to reduced row-echelon form, we obtain

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

000000001000010100010011

The corresponding system of equations is

x1 + x2 + x5 = 0x3 + x5 = 0

x4 = 0

Thus the general solution isx1 = -s – t, x2 = s, x3 = -t, x4 = 0, x5 = t,

where s and t are parameters.

Note: The trivial solution is obtained when s = t = 0.

Theorem

A homogeneous system of linear equations with more unknowns than equations has infinitely many

solutions.

Problem 1Solve the following system of nonlinear equations for x, y and z.

x2 + y2 + z2 = 6x2 - y2 + 2z2 = 22x2 + y2 - z2 = 3

Problem 2Show that the following nonlinear system has eighteen solutions if 0 ≤ α ≤ 2π, 0 ≤ β ≤ 2π , 0 ≤ γ ≤ 2 π.

sin α + 2 cos β + 3 tan γ = 02 sin α + 5 cos β + 3 tan γ = 0- sin α - 5 cos β + 5 tan γ = 0

Matrices and matrix Operations

Matrices and matrix Operations

By: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 followingrectangular 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.

DefinitionTwo matrices are defined to be equal if the have the same size and their corresponding entries are equal.

DefinitionIf A and B are matrices of the same size, then the sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A.

DefinitionIf A is any matrix and c is any scalar, then the product cAis the matrix obtained by multiplying each entry of A by c.

DefinitionIf 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

DefinitionIf A is any m x n matrix, then the transpose of A, denoted by AT, is defined to be the n x m matrix that results from interchanging the rows and columns of A, that is, the first column of AT is the first row of A, the second column of AT is the second row of A, and so forth.

DefinitionIf A is a square matrix, then the trace of A, denoted by tr(A), is defined to be the sum of the entries on the main diagonal of A. The trace of A is undefined if A is not a square matrix.

DefinitionIf A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, where I is an identity matrix, then A is said to be invertible and B is called an inverse of A.

TheoremIf B and C are both inverses of the matrix A, then B = C.

TheoremIf A and B are invertible matrices of the same size, then AB is invertible and (AB)-1 = B-1A-1.

Theorem

If A is an invertible matrix, thena). A-1 is invertible and (A-1)-1 = Ab). An is invertible and (An)-1 = (A-1)n for n = 1,2, …c). For any nonzero scalar k, the matrix kA is invertible and

(kA)-1 = (1/k) A-1.d). AT is invertible and (AT)-1 = (A-1)T.

Proof: Exercise for student.

Problem 1

A square matrix A is called symmetric if AT = A and skew-symmetric if AT = -A. Show that if B is a square matrix, thena). BBT and B + BT are symmetric.

b). B – BT is skew-symmetric.

Problem 2

Let A be a square matrix. a). Show that (I – A)-1 = I + A + A2 + A3 + A4 if A5 = 0.b). Show that (I – A)-1 = I + A + A2 + … + An if An+1 = 0.

Elementary Matrices and a Method for Finding A-1

DefinitionAn n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix In by performing a single elementary row operation.

TheoremIf the elementary matrix A results from performing a certain row operation on Im and if A an m x n matrix, then the product EA is the matrix that results when this same row operation is performed on A.

SOME EXAMPLES OF ELEMENTARY MATRICES

EXAMPLE :

TheoremEvery elementary matrix is invertible, and the inverse is also an elementary matrix.

TheoremIf 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 In.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 In to obtain A-1.

Example: Find the inverse of

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

801352321

A

Solution :

Thus,