ENGG2013 Unit 93x3 Determinant

Feb, 2011.

Last time

• 22 determinant• Compute the area of a parallelogram by

determinant• A formula for 2x2 matrix inverse

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Today

• 33 determinant and its properties• Using determinant, we can

– test whether three vectors lie on the same plane– solve 33 linear system– test whether the inverse of a 33 matrix exists

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Vector Notation

• We will use two different notations for a point in the 3D space

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(x,y,z)

x

z

y

x

z

y

22 determinant

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Notation for 22 determinant :

How to calculate: ad – bc

ad

+

- bc

–

33 determinant

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Notation for 33 determinant :

Definition:

Rule of Sarrus

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+ + + – – –

Pierre Frédéric Sarrus (1798 – 1861)

Volume of parallelepiped

• Geometric meaning– The magnitude of 33 determinant is the

volume of a parallelepiped

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z

y

x

Co-planar zero determinant

• Determinant = 0 Volume = 0

the three vectors lie on the same plane

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z

y

x

A collection of vectorsare said to be co-planarif they lie on the same plane.

Det of Diagonal matrix

• Volume of a rectangular box

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a

b

c

Transpose has the same determinant

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+ + + – – –

Compare with

Volume of parallelepiped

• In computing the volume of a parallelepiped, it does not matter whether we write the vector horizontally or vertically in the determinant

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Volume of parallelepiped with vertices(0,0,0), (1,2,0), (2,0,1), (–1, 1, 3) equals tothe absolute value of

or

x

y

z

Question

• Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on the same plane?

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Cramer’s rule

• If the determinant of a 33 matrix A is non-zero, we can solve the linear system A x = b by Cramer’s rule.

• The solution to

is

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Gabriel Cramer (1704-1752)

or equivalently

A bx

PROPERTIES OF DETERMINANT

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How to show that Cramer’s rule does give the correct answer?

• The Cramer’s rule is a theorem, which requires a proof, or verification.

• We need some properties of determinant.

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Properties of determinant

1. Taking transpose does not change the value of determinant

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We have already verified this property in p.11

Meta-property

• Because1. After taking the transpose of a matrix, columns

become rows, and rows become column.2. Taking the transpose of a matrix does not

change the value of its determinant.

• Therefore, any row property of determinant is automatically a column property, and vice versa.

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Properties of determinant

2. If any row or column is zero, then the determinant is 0.

For example

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Properties of determinant3. If any two columns (or rows) are the identical, then the determinant is zero.

For example, if the second column and the third column are the same, then

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Properties of Determinant

4. If we exchange of the two columns (or two rows), the determinant is multiplied by –1.

For example, if we exchange the column 2 and column 3, we have

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The first kindof elementaryrow operation

Multiply by a constant

5. If we multiply a row (or a column) by a constant k, the value of determinant increases by a factor of k.

For example, if we multiply the third row by a constant k,

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The 2nd kindof elementaryrow operation

An additive property

6. If a row (or column) of a determinant is the sum of two rows (or columns), the determinant can be split as the sum of two determinants

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For example, if the first column is the sum of two column vectors, thenwe have

Properties of Determinant

7. If we add a constant multiple of a row (column) to the other row (column), the determinant does not change.

For example, if we replace the 3rd column by the sum of the 3rd column and k times the 2nd column,

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The 3rd kindof elementaryrow operation

Summary on the effect of theelementary row (or column) operations on determinant

• Exchange two rows (or columns) change the sign of determinant

• Multiply a row (or a column) by a constant k multiply the determinant by k

• Add a constant multiple of a row (column) to another row (or column) no change

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Proof of the Cramer’s rule• The solution to

is

Verification for x1: Substitute the value of b1, b2 and b3 in the first column of A.

Verification for x2: Substitute the value of b1, b2 and b3 in the second column of A.

Etc.

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Cramer’s rule in wikipedia

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Because x1, x2, x3 satisfy the system of linear equations, we have

By substitution

Property 6

Property 5

=0 =0 By Property 3

MINOR AND COFACTOR

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Another way to compute det

Group the six terms as

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33 determinant can be computed in terms of 22 determinant

Minor and cofactor

• A minor of a matrix is the determinant of some smaller square matrix, obtained by removing one or more of its rows and columns.

• Notation: Given a matrix A, the minor obtained by removing the i-th row and j-th column is denoted by Aij. It is also called the minor of the (i,j)-entry aij in A.

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Expansion on the first row

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Minor of a1 Minor of b1 Minor of c1

Expansion on the second row

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Minor of a2 Minor of b2 Minor of c2

Expansion on the third row

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Minor of a3 Minor of b3 Minor of c3

The sign pattern

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Expansion on the first row

Expansion on the second row

Expansion on the last row

Column expansion

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For example,

We have similar recursive formula fordeterminant by column expansion

Computation on the third column is easy, because there are lots of zeros.

Cofactor

• The minor together with the appropriate sign is called cofactor.

• For

The cofactor of Cij is defined as

Expansion on the i-th row (i=1,2,3):

Expansion on the j-th column (j=1,2,3):

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The minor of aij

The sign

A formula for matrix inverse

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Suppose that det A is nonzero.

Three steps in computing above formula1. for i,j = 1,2,3, replace each aij by cofactor Cij

2. Take the transpose of the resulting matrix.3. divide by the determinant of A.

Usually called the adjoint of A

(Beware of thesubscripts)

A Quotation

Algebra is but written geometry;

geometry is but drawn algebra.--- Sophie Germain (1776-1831)

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L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée