Systems of Linear Equations

Systems of Linear

Equations

Systems of Linear

Equations

By:

Tri Atmojo Kusmayadi and Mardiyana Mathematics Education

Standard of Competency:

Understanding the properties of systems of linear equations,

Basic Competencies:

3

1

A. Introduction to Systems of Linear Equations

Definition

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

a₁x₁ + a₂x₂ + … + a_nx_n = b

Definition

A solution of a linear equation a₁x₁ + a₂x₂ + … + a_nx_n = b is a sequence of n numbers s₁, s₂, …, s_n such that the equation is satisfied when we substitute x₁ = s₁, x₂ = s₂, …, x_n = s_n.

The set of all solutions of the equation is called its

Examples :

„

Find the solution set of (a)

and (b)

„

Solutions : (a)

Linear equations Systems

A finite set of linear equations in the variables x₁, x₂, …, x_n is called a system of linear equations or a linear systems.

„

For example, the system

„

has the solution

„

However,

Every system of linear equations has no

solutions, or has exactly one solution, or has

infinitely many solutions.

The lines l₁and l₂ may be

parallel, in which case there is no intersection and consequently no solution to the system.

The lines l₁and l₂ may intersect at only one point, in which case the system has exactly one solution.

The General Form of Linear Systems

An arbitrary system of m linear equations in n variables can be written as

a₁₁x₁ + a₁₂x₂ + … + a_1nx_n = b₁ a₂₁x₁ + a₂₂x₂ + … + a_2nx_n = b₂

: : : :

a_m1x₁ + a_m2x₂ + … + a_mnx_n = b_m

Augmented Matrices

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

⎥

„ _{For example, the augmented matrix for the system of}

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

operating on the equations in the systems

x + y + 2z = 9 2x + 4y – 3z = 1 3x + 6y – 5z = 0

In the following column below we solve a system of

linear equations by

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

⎥

equation to the second equation and add -3 times the first equation to the third equation to obtain

Multiply the second row by ½ Multiply the second equation by ½

to obtain

x + y + 2z = 9 y – 7/2 z = - 17/2 3y – 11z = -27

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

x + y + 2z = 9 y – 7/2 z = - 17/2

-- ½ z = -3/2

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

Multiply the third row by-2 to Multiply the third equation by -2 to

obtain

x + y + 2z = 9 y – 7/2 z = - 17/2

z = 3

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

x + 11/2 z = 35/2 y – 7/2 z = - 17/2

z = 3

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

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

Thus the solution of the system of linear equations is

Gaussian

Elimination

Gaussian

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.

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 + … + a_1nx_n = 0 a₂₁x₁ + a₂₂x₂ + … + a_2nx_n = 0 : : : : a_m1x₁ + a_m2x₂+ … + a_mnx_n = 0

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

Example

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

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

Solution:

The augmented matrix for the system is

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

The corresponding system of equations is

x₁ + x₂ + x₅ = 0 x₃ + x₅ = 0

x₄ = 0

Thus the general solution is

x₁ = -s – t, x₂ = s, x₃ = -t, x₄ = 0, x₅ = t, where s and t are parameters.

Theorem

A homogeneous system of linear equations with

more unknowns than equations has infinitely many

Problem 1

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

x2 _{+ y}2 _{+ z}2 _{= 6}

x2 _{- y}2 _{+ 2z}2 _{= 2}

2x2 _{+ y}2 _{- z}2 _{= 3}

Problem 2

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

Matrices and matrix

Operations

Matrices and matrix

Operations

By:

In the previous Section we used rectangular arrays of numbers, called augmented matrices, to abbreviate systems of linear equations.

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

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

Definition

I f A is any m x n m at rix, t hen t he t r a n spose of A, denot ed by AT_{, 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 colum ns of A, t hat is, t he first colum n of AT _{is t he}

first row of A, t he second colum n of AT _{is t he}

second row of A, and so fort h.

Definition

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 a). A-1 _{is invertible and (A}-1₎-1 _{= A}

b). An _{is invertible and (A}n₎-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 (A}T₎-1 _{= (A}-1₎T_.

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, then}

a). BBT _{and B + B}T _{are symmetric.}

b). B – BT _{is skew-symmetric.}

Problem 2

Let A be a square matrix.

a). Show that (I – A)-1 _{= I + A + A}2 _{+ A}3 _{+ A}4 _{if A}5 _{= 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

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.

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