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MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

1.Systems of Equations and Matrices

1.1.Representation of a linear system.The general system of m equations in n unknowns can

be written a

11x1+a12x2+···+a1nxn=b1

a

21x1+a22x2+···+a2nxn=b2

a

31x1+a32x2+···+a3nxn=b3

a m1x1+am2x2+···+amnxn=bm(1)

In this system, a

ijis a given real number as is bi. Specifically, aijis the coefficient for the unknown x jin the ithequation. We call the set of all the aijarranged in a rectangular array the coefficient matrix of the system. Using matrix notation we canwrite the system as Ax=b (a

11a12···a1n

a

21a22···a2n

a

31a32···a3n............

a m1am2···amn))))))) x 1 x 2... x n))))) =(((((b 1 b 2... b m))))) (2) We define the augmented coefficient matrix for the system as

A=(((((((a

11a12···a1nb1

a

21a22···a2nb2

a

31a32···a3nb3............

a m1am2···amnbm))))))) (3)

Consider the following matrix A and vector b

A=(( 1 2 1 2 5 2 -3-4-2)) , b=(( 3 8 -4)) (4)

We can then write

Date: 26 March 2008.

1

2 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

Ax=b (1 2 12 5 2 -3-4-2)) (x 1 x 2 x 3)) 3 8 -4)) (5) for the linear equation system x

1+ 2x2+x3= 3

2x1+ 5x2+ 2x3= 8

-3x1-4x2-2x3=-4(6)

1.2.Row-echelon form of a matrix.

1.2.1.Leading zeroes in the row of a matrix.A row of a matrix is said to have k leading zeroes if

the first k elements of the row are all zeroes and the (k+1) stelement of the row is not zero.

1.2.2.Row echelon form of a matrix.A matrix is in row echelon form if each row has more leading

zeroes than the row preceding it.

1.2.3.Examples of row echelon matrices.The following matrices are all in row echelon form

A=(( 3 4 7 0 5 2

0 0 4))

B=(( 1 0 1 0 0 2

0 0 0))

C=(( 1 3 1 0 4 1

0 0 3))(7)

The matrix C would imply the equation system

x

1+ 3x2+x3=b1

4x2+x3=b2

3x3=b3

which could be easily solved for the various variable after dividing the third equation by 3.

1.2.4.Pivots.The first non-zero element in each row of a matrix in row-echelon form is called a

pivot. For the matrix A above the pivots are 3,5,4. For the matrix B they are 1,2 and for C they are 1,4,3. For the matrix B there is no pivot in the last row.

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 3

1.2.5.Reduced row echelon form.A row echelon matrix in which each pivot is a 1 and in which each

column containing a pivot contains no other nonzero entries, is said to be in reduced row echelon form. This implies that columns containing pivots are columns of an identity matrix. In the matrix D in equation (8) all three columns are pivot columns. In the matrix E in equation (8) all the first two columns are pivot columns. The matrices D and E are both inreduced row echelon form. D=(( 1 0 0 0 1 0

0 0 1))

E=(( 1 0 0 0 1 0

0 0 0))

(8) In matrix F in equation (9), columns 2, 4 and 5 are pivot columns. The matrix F is in row echelon form but not in reduced row echelon form because of the 1 in thea25and the 3 in the a15elements of the matrix. The 5 in the a

13element is irrelevant because column 3 is not a pivot column.

F=((((0 1 5 0 30 0 0 1 10 0 0 0 10 0 0 0 0))))

(9)

1.2.6.Reducing a matrix to row echelon or reduced row echelon form.Any system of equations can

be reduced to a system with the A portion of the augmented matrix˜Ain echelon or reduced echelon

form by performing what are called elementary row operations on the matrix˜A. These operations are

the same ones that we used when solving a linear system using the method of Gaussian elimination. There are three types ofelementary row operations. Performing these operations does not change the basic nature of the system or its solution.

1:Changing the order in which the equations or rows are listed produces an equivalent system.

2:Multiplying both sides of a single equation of the system (ora single row of˜A) by a nonzero

number (leaving the other equations unchanged) results in asystem of equations that is equivalent to the original system.

3:Adding a multiple of one equation or row to another equation or row (leaving the other

equations unchanged) results in a system of equations that is equivalent to the original system.

1.2.7.Rank.

Definition 1(Rank).The number of non-zero rows in the row echelon form of an m×n matrix A produced by elementary operations on A is called the rank of A. Alternatively, the rank of an m× n matrix A is the number of pivots we obtain by reducing a matrix to row echelon form. Matrix D in equation 8 has rank 3, matrix E has rank 2, while matrix F in 9 has rank 3. Definition 2(Full column rank).We say that an m×n matrix A has full column rank if r = n.

If r = n, all columns of A are pivot columns.

The matrix L is of full column rank

L=(( 1 0 0 1 0 0))

4 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONSDefinition 3(Full row rank).We say that an m×n matrix A has full row rank if r = m. If r =

m, all rows of A have pivots and the reduced row echelon form has no zero rows.

The matrix M is of full row rank

M=((

1 0 0-10

0 1 0 4

0 0 1-7))

1.2.8.Solutions to equations (stated without proof).

a. A system of linear equations with coefficient matrix A whichis m×n, a right hand side vector b which is m×1, and augmented matrixˆAhas a solution if and only if rank(A) =rank(ˆA) This means that we can write the b vector as a linear combination of the columns of A. b. A linear system of equations must have either no solution,one solution, or infinitely many solu- tions. c. If the rank of A = r = n1, b2, ... , bm, if and only if number of rows of A=number of columns of A=rank(A)

1.3.Solving systems of linear equations by finding the reduced echelon form of a matrix

and back substitution.To solve a system of linear equations represented by a matrixequation, we

first add the right hand side vector to the coefficient matrix toform the augmented coefficient matrix.

We then perform elementary row and column operations on the augmented coefficient matrix until it is in row echelon form or reduced row echelon form. If the matrix is in row echelon we can solve

it by back substitution, if it is in reduced row echelon form we can read the coefficients off directly

from the matrix. Consider matrix equation 5 with its augmented matrix˜A. Ax=b (1 2 12 5 2 -3-4-2)) (x 1 x 2 x 3)) 3 8 -4)) A=((

1 2 1 3

2 5 2 8

-3-4-2-4))(5)

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 5

Step 1. Consider the first row and the first column of the matrix. Element a11is a non-zero number and so is appropriate for the first row of the row echelon form.If element a11were zero, we would exchange row 1 with a row of the matrix with a non-zeroelement in column 1.

Step 2. Use row operations to "knock out" a

21by turning it into a zero. This would then make

the second row have one more leading zero than the first row. Wecan turn the a21element into a zero by adding negative 2 times the first row to the second row. -2×?1 2 1 3? ?-2-4-2-6? -2-4-2-6

2 5 2 8

0 1 0 2

This will give a new matrix on which to operate. Call it

˜A1.

A1=((

1 2 1 3

0 1 0 2

-3-4-2-4)) (10)

We can obtain

˜A1by premultiplying˜Aby what is called the a21elimination matrix denoted by E

21where E21is given by

E 21=((
1 0 0 -2 1 0

0 0 1))

(11)

The first and third rows of E

21are just rows of the identity matrix because the first and

third rows of˜Aand˜A1are the same. The second row of˜A1is a linear combination of the

rows of˜Awhere we add negative 2 times the first row of˜Ato 1 times the second row of˜A. To convince yourself of this fact remember what it means to premultiply a matrix by a

row vector. In the following example premultiplying the matrix((1 22 5 -3-4)) by the vector ?1 0 2?adds one times the first row to twice the third row as follows.

1 0 2?((1 22 5

-3-4)) =?-5-6?.

For the case at hand we have

A1=(( 1 0 0 -2 1 0

0 0 1))

(1 2 1 32 5 2 8 -3-4-2-4))

1 2 1 3

0 1 0 2

-3-4-2-4)) (12)

Step 3. Use row operations to "knock out" a

31by turning it into a zero. This would then make the

third row have one more leading zero than the first row. We can turn the a31element into a zero by adding 3 times the first row to the third row.

6 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

3×?1 2 1 3?

?3 6 3 9?

3 6 3 9

-3-4-2-4

0 2 1 5

This will give a new matrix on which to operate. Call it

˜A2.

A2=((

1 2 1 3

0 1 0 2

0 2 1 5))

(13)

We can obtain

˜A2by premultiplying˜A1by what is called the a31elimination matrix denoted by E

31where E31is given by

E 31=((
1 0 0 0 1 0quotesdbs_dbs20.pdfusesText_26

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