•Introduction to System of Linear Equations

• Gaussian Elimination

• Matrices and Matrix Operations

• Inverses; Rules of Matrix Arithmetic

• Elementary Matrices and a Method for Finding

• Further Results on Systems of Equations and Invertibility

• Diagonal, Triangular, and Symmetric Matrices

Chapter 1 Systems of Linear Equations and Matrices

1A

1.1 Introduction to systems of linear equations

•Any straight line in xy-plane can be represented algebraically by an equation of the form:

1 2a x a y b

1 2, ,..., nx x x

1 1 2 2 ... n na x a x a x b

•General form: Define a linear equation in the n variables :

where a1, a2, …, an and b are real constants.

The variables in a linear equation are sometimes called unknowns.

Examples:

• Some linear equations:

1 2 3 43 5 6 0,109 123 100 98, 2 4 5 9x y z x y x x x x

2 6 0, 9 3 2 , 9, 3ln 4 8, cos tan 6xx y xz x y e y y x z x y

Observe that

•A linear equation does not involve any products or roots of variables

• Some NON linear equations:

•All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions.

Linear Equations

•A solution of a linear equation is a sequence of n numbers s1, s2, …, sn such that the equation is satisfied.

Example:

(a) Find the solution of 3x+4y = 5

•The set of all solutions of the equation is called its solution set or general solution of the equation.

Solutions of Linear Equations

Example (b): Find the solution of 1 2 33 8 16x x x

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

1 1 1 1 1

2 2 2 2 2

( , )

( , )

a x b y c a b not both zero

a x b y c a b not both zero

A sequence of numbers s1, s2, …, sn is called a solution of the system if every equation is satisfied.

A system has no solution is said to be inconsistent.

If there is at least one solution of the system, it is called consistent.

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

A general system of two linear equations:

Two line may be parallel – no solution Two line may be intersect at only one point – one solution Two line may coincide – infinitely many solutions

Linear Systems

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

...

...

...

...

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

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

Where x1, x2, …, xn are unknowns and the subscripted a’s and b’s denote Constants.

Note that is in the ith equation and multiplies unknown ija jx

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

...

...

...

...

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

11 12 1 1

21 22 2 2

1 2

...

...

... ... ... ... ...

...

n

n

m m mn m

a a a b

a a a b

a a a b

A system of m linear equations in n unknowns can be written as a rectangular array of numbers:

This is called the augmented matrix for the system.

Example:

1 2 3

1 2 3

1 2

2 3 5

6 11 4 0

5 9 2 1n

x x x

x x x

x x x

Augmented Matrix

Since the rows of an augmented matrix correspond to the equations, we can apply the following three types of operations to solve systems of linear equations.

These operations are called elementary row operations.

We will go to details in the next session.

•Multiply an equation through by an nonzero constant

• Interchange two equation

• Add a multiple of one equation to another

Elementary Row Operations

1.2 Gaussian Elimination

A matrix which has the following properties is in reduced row echelon Form.

•If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leader 1.

•If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.

•In any two successive rows that do not consist entirely of zeros, the leader 1 in the lower row occurs farther to the right than the leader 1 in the higher row.

•Each column that contains a leader 1 has zeros everywhere else.

A matrix that has the first three properties is said to be in row echelon form.

Note: A matrix in reduced row-echelon form is of necessity in row echelonform, but not conversely

Example: Determine which of the following matrices are in row-echelon form, reducedRow-echelon form, both, or neither.

1 0 0 1 8 2 1 0 0

0 1 0 , 0 1 0 , 0 3 0

0 0 1 0 0 0 0 2 1

1 0 0 2 0 1 5 4 1 5 4 2

0 1 0 3 , 0 1 0 7 , 0 1 6 1

0 0 1 3 0 0 0 1 0 0 0 0

Solution:

Row-Echelon and Reduced Row-Echelon Forms

Example: Determine which of the following matrices are in row-echelon form, reducedRow-echelon form, both, or neither.

1 0 0 1 8 2 1 0 0

0 1 0 , 0 1 0 , 0 3 0

0 0 1 0 0 0 0 2 1

1 0 0 2 0 1 5 4 1 5 4 2

0 1 0 3 , 0 1 0 7 , 0 1 6 1

0 0 1 3 0 0 0 1 0 0 0 0

Solution:

All matrices of the following types are in row-echelon form (any real numbers substituted for the *’s. ) :

All matrices of the following types are in reduced row-echelon form (any real numbers substituted for the *’s. ) :

0 1 * * * * * * * *1 * * * 1 * * * 1 * * *

0 0 0 1 * * * * * *0 1 * * 0 1 * * 0 1 * *

, , , 0 0 0 0 1 * * * * *0 0 1 * 0 0 1 * 0 0 0 0

0 0 0 0 0 1 * * * *0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 *

0 1 * 0 0 0 * * 0 *1 0 0 0 1 0 0 * 1 0 * *

0 0 0 1 0 0 * * 0 *0 1 0 0 0 1 0 * 0 1 * *

, , , 0 0 0 0 1 0 * * 0 *0 0 1 0 0 0 1 * 0 0 0 0

0 0 0 0 0 1 * * 0 *0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 *

More on Row-Echelon and Reduced Row-Echelon Form

Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form. Solve the system.

1 0 0 6 1 0 0 5 6

( ) 0 1 0 3 , ( ) 0 1 0 1 3

0 0 1 8 0 0 1 7 9

1 3 0 0 2 81 0 0 0

0 0 1 0 3 5( ) , ( ) 0 1 4 0

0 0 0 1 4 70 0 0 1

0 0 0 0 0 0

a b

c d

Solution (a):

Solutions of Linear Equations

Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.Solve the system. 1 0 0 5 6

( ) 0 1 0 1 3

0 0 1 7 9

b

Solution (b):

Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.Solve the system.

1 3 0 0 2 8

0 0 1 0 3 5( )

0 0 0 1 4 7

0 0 0 0 0 0

c

Solution (c):

Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.Solve the system.

1 0 0 0

( ) 0 1 4 0

0 0 0 1

d

Solution (d):

Elimination Method

Elimination procedure to reduce a matrix to reduced row-echelon form.

0 0 2 0 7 12

2 4 10 6 12 28

2 4 5 6 5 1

Step 1. Locate the leftmost column that does not consist entirely of zeros.

Step 2. Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column found in step 1.

0 0 2 0 7 12

2 4 10 6 12 28

2 4 5 6 5 1

2 4 10 6 12 28

0 0 2 0 7 12

2 4 5 6 5 1

R1 R2

Step 3. If the entry that is now at the top of the column found in step 1 is a, multiplyThe first row by 1/a in order to introduce a leading 1.

1 2 5 3 6 14

0 0 2 0 7 12

2 4 5 6 5 1

1

1

2R

Step 4. Add suitable multiples of the top row to the rows below so that all entriesBelow the leading 1 become zeros.

1 2 5 3 6 14

0 0 2 0 7 12

0 0 5 0 17 29

3 1( 2 )R R

Step 5. Cover the top row in the matrix and begin again with Step 1 applied to theSubmatrix that remains. Continue in this way until the entire matrix is in row-echelon form.

1 2 5 3 6 14

0 0 1 0 7 / 2 6

0 0 5 0 17 29

2

1

2R

3 15R R1 2 5 3 6 14

0 0 1 0 7 / 2 6

0 0 0 0 1/ 2 1

1 2 5 3 6 14

0 0 1 0 7 / 2 6

0 0 0 0 1 2

32R

Step 6. Beginning with the last nonzero row and working upward, add suitable multiples of each row to the rows above to introduce zeros above the leading 1’s

1 2 5 3 6 14

0 0 1 0 7 / 2 6

0 0 0 0 1 2

2 3

7

2R R 1 2 5 3 6 14

0 0 1 0 0 1

0 0 0 0 1 2

1 36R R 1 2 5 3 0 2

0 0 1 0 0 1

0 0 0 0 1 2

1 2 0 3 0 7

0 0 1 0 0 1

0 0 0 0 1 2

1 25R R

•Step1~Step5: the above procedure produces a row-echelon form andis called Gaussian elimination

•Step1~Step6: the above procedure produces a reduced row-echelonform and is called Gaussian-Jordan elimination

Every matrix has a unique reduced row-echelon form but a row-echelonform of a given matrix is not unique

Example: Solve by Gauss-Jordan elimination2 9

2 4 3 1

3 6 5 0

x y z

x y z

x y z

Solution:

A system of linear equations is said to be homogeneous if the constantterms are all zero; that is, the system has the form:

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

There are only two possibilities for its solutions:• There is only the trivial solution

• There are infinitely many solutions in addition to the trivial solution

11 1 12 2 1

21 1 22 2 2

1 1 2 2

... 0

... 0

...

... 0

n n

n n

m m mn n

a x a x a x

a x a x a x

a x a x a x

Homogeneous Linear Systems

Example: Solve the homogeneous system of linear equations by Gauss-Jordan elimination

1 2 3 5

1 2 3 4 5

1 2 3 5

3 4 5

2 +2 + 0

2 -3x 0

2 0

0

x x x x

x x x x

x x x x

x x x

Solution:

Theorem 1.2.1A homogeneous system of linear equations with more unknownsthan equations has infinitely many solutions.

Remark•This theorem applies only to homogeneous system!

• A nonhomogeneous system with more unknowns than equations need not be consistent; however, if the system is consistent, it will have infinitely many solutions.