DETERMINANT MATRIX YULVI ZAIKA - PowerPoint Presentation

Slide1

DETERMINANT MATRIX

YULVI ZAIKASlide2

DETERMINANT

a “determinant” is a certain

kind of

function that associates a real number with a square matrixWe will obtain a formula for the inverse of an invertible matrix as well as a formula for the solution to certain systems of linear equations in terms of determinants.is invertible if ad-bc 0 . The expression ad – bc occurs so frequently in mathematics that it has a name; it is called the determinant of the matrix A and is denoted by the symbol det A or |A| . With this notation, the formula A-1for given inSlide3

Finding Minors and CofactorsSlide4

The definition

3x3 of

a determinant in terms of minors and cofactors isSlide5

Determinant matrix 3x3Slide6

Cofactor Expansion Along the First ColumnSlide7

Adjoint of matrix

If

A is any

nxn matrix and Cij is the cofactor of aij, then the matrixis called the matrix of cofactors from A

. The transpose of this matrix is called the adjoint

of A

and is denoted by

adj

A.Slide8

EX

If

is A

an invertible matrix, thenSlide9

If A

is

an

nxn triangular matrix (upper triangular, lower triangular, or diagonal), then det(A) is the product of theentries on the main diagonal of the matrix; that is, det (A)= a11. a22a33…ann.Determinant of an Upper Triangular MatrixSlide10

Cramer's Rule

If Ax=b is a

system of linear equations in unknowns such

that det (A) 0 , then the system has a unique solution. This solution isWhere Aj is the matrix obtained by replacing the entries in the

j th

column of

A

by the entries

in the

matrixSlide11
Slide12
Slide13

Use Cramer's rule to solve Slide14

EVALUATING DETERMINANTS

BY

ROW REDUCTION

Let A be a square matrix. If A has a row of zeros or a column of zeros, then det A=0 .Let A be a square matrix. Then det A= det AT .Let A be an nxn

matrix(a) If B is the matrix that results when a single row or single column of A is multiplied by a scalar ,

then

det

(A)=k

det

(B)

.

(b)

If

B is

the matrix that results when two rows or two columns of

A are

interchanged,

then

det

(B)=-

det

(A)

If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column

is added

to another column,

then

det

(B)=

det

(A)

.Slide15
Slide16

Elementary Matrices

Let

E be an

nxn elementary matrix.If E results from multiplying a row of In by k, then det E=k. If E results from interchanging two rows of In , then det(E)=-1 .If E results from adding a multiple of one row of to another, then det E=1.Slide17

Matrices with Proportional Rows or Columns

If A is a square matrix with two proportional rows or two proportional columns,

then

det (A)=0 .Slide18

Evaluating Determinants by Row ReductionSlide19

This determinant could be computed as above by using elementary row operations to reduce

A

to row-echelon form, but we can put

A in lower triangular form in one step by adding-3 times the first column to the fourth to obtainSlide20

Row Operations and Cofactor Expansion

By adding suitable multiples

of

the second row to the remaining rows, we obtainSlide21

PROPERTIES OF

THE DETERMINANT FUNCTIONSlide22

Linear Systems of the Form Ax=

x

Many applications of linear algebra are concerned with systems

of n linear equations in n unknowns that are expressed in the formAx=xwhere is a scalar. Such systems are really homogeneous linear systems in disguise, since the equation can be rewritten as x-Ax=0 or, by inserting an identity matrix and factoring, as(I-A)x=0Slide23

Finding

I-A



Is called

a characteristic value

or an

eigenvalue

*

of A and

the nontrivial solutions of

eq

are called the

eigenvectors

of

A

corresponding to .Slide24

The factored form of this equation is

(

+2)(-5)

, so the eigenvalues of A are -2 and 5.Jika

=-2

Jika

=5