nghiên cứu khoa học giảng dạy và học tập

Expand according to the fourth column.. Any modern microcomputer can calculate a 25 x 25 determinant in a fraction of a second, since less than 5300 such operations are required.[r]

I – The Determinant of Matrix

II – Properties of Determinant

III – Laplace’s Expansion

Let be a square matrix

Determinant of A is denoted by det

Let denote the submatrix formed by deleting the i th row and

j th column of A;

(i,j)- cofactor:

Definition of (i,j)- cofactor

Compute det (A), where

Example

Solution

The determinant of an matrix A may be computed by a cofactor expansion along any row or down any column

Compute det (A), where

Example

We expand across the 3th row

Solution

Compute det (A), where

Example

We expand down the 2th column

Solution

If A is a triangular matrix, then det(A) is the product of the entries on the main diagonal of A.

Example

Using Row Operation for Calculating Determinant

a If a multiple of one row of A is added to another to produced a

matrix B, then det(B) = det(A).

b If two row of A interchanged to produced B, then det(B) = - det(A).

c If one row of A is multiplied by scalar k to produced a B, then

det(B) =k det(A).

Triangular matrix

B

Using Row Operation for Calculating Determinant

1 If then

2 If then

3 If then

Find the determinant of a matrix A, using elementary operations

Expand according to the first column

Solution

Step 1 Select one column (or one row) of the matrix

Step 2 Choose one nonzero element of the selected column (or selected row) Using row ( or column) operations to eliminate all others elements except selected

Step3 Expand the determinant according to the selected row ( or column)

The Formula of Calculating Determinant

Find the determinant of a matrix A, using elementary operations

Expand according to the fourth column

A cofactor expansion requires over n! multiplications.

If a supercomputer could make one trillion multiplications per second, it would have to run for over 500.000 years to compute a

25x25 determinant by cofactor expansion (required 25! is approximately 1.5x1025 operations)

Most computer programs that compute det (A) using a row

operations

The row operations requires (n 3 +2n-3)/3 multiplications and

divisions Any modern microcomputer can calculate a 25x25 determinant in a fraction of a second, since less than 5300 such operations are required

If A is an nxn matrix, then det (AT) = det (A)

det(AB) = det(A) det(B)

If a matrix A has a zero row, then det (A) = 0

If a matrix A has two identical rows, then det (A) = 0

Let A be an invertible nxn matrix There exists an inverse

A-1, such that AA-1 = I It follows that

, where

Proof

A square matrix A is invertible if and only if det(A) 0

Theorem

Suppose that det(A) 0 Then

det(AA-1) = det (I) det(A).det(A-1) = 1 det(A) 0

Let A be an invertible nxn matrix Then

, where

Determinant Formula for A-1

Exp Find the inverse of the matrix

The nine cofactors are

Properties of an invertible matrix

1

2 If A is invertible, then

Proof

Suppose k is any natural number smaller than n and i1, i2, …, ik and j1,

j2, …, jk are arbitrary numbers satisfying the conditions

-The sub-determinant of order k, denoted by , is determinant of

order k corresponding to the matrix formed by the elements of matrix A lying at the intersection of k rows labeled i1, i2, …, ik and k columns labeled j1, j2, …, jk

Definition of a sub-determinant of order k

The k-minor of the sub determinant of order k is

determinant of order n - k corresponding to the matrix obtained from A

by deleting k rows labeled i1, i2, …, ik and k columns labeled j1, j2, …, jk

Definition of k-Minor

-Theorem (Laplace’s Expansion)

For any natural number k smaller than n and for any fixed numbers of rows i1, i2, …, ik such that , the following formula holds true

This formula is called the expansion of the determinant according to k rows i1, i2, …, ik The summation in this formula is carried out by all possible values of the indeces j1, j2, …, jk satisfying the conditions

The quantity

is called a k – cofactor of

-Example

Find the determinant of a matrix A, using Laplace’s Expansion

-Solution

Select k = 2 and select 2 rows: the second and the fourth rows

There are kxk submatrices of the first type, but only one nonzero

Calculate det(A), where

Calculate det(A), where

Which one of the following statements is true?

a) The degree of polynomial f(x) is 5.

b) The degree of polynomial f(x) is 4.

c) The degree of polynomial f(x) is 3.

d) The others statements are false

Calculate the determinant of the following matrix

Calculate the determinant

Solve the equation, where a, b, c are real numbers.

Solve the equation

Find the determinant

Calculate

Find

Compute

Solve the following equation in C

Calculate det(A) using Laplace’s Expansion, where

Find an inverse of the following matrix A, using the Determinant’s formula

Find an inverse of the following matrix A, using the Determinant’s formula

Find all m such that a matrix A is invertible.

Find all m such that a matrix A is invertible.

Let 1) Calculate det (A-1).

2) Calculate det (5A)-1 3) Calculate det (PA)

Let

1) Find det (4AB)-1 2) Find det (PAB)