ENGG2013 Unit 82x2 Determinant

Jan, 2011.

Last time

• Invertible matrix (a.k.a. non-singular matrix)– Represents reversible linear transformation

• Gauss-Jordan elimination– Algorithm for compute matrix inverse

kshum ENGG2013 2

Carl Friedrich Gauss (1777 – 1853)

• Borned: Braunschweig, Germany.• Died: Göttingen, Germany.• Great mathematician in the 18th century.• Legacy

• Gaussian distribution• Gaussian elimination• Gauss-Jordan elimination• Gaussian curvature in differential geometry• Proof of the quadratic reciprocity in number

theory• Construction of 17-gon by straightedge and

compass• And much more

kshum ENGG2013 3

Wilhelm Jordan (1842 –1899)

• Borned: Ellwangen, Germany.• Died: Hanover, Germany.• Geodesist.• Remembered for:

– Surveying in Germany and Africa

– His book “Textbook of Geodesy” (Handbuch der Vermessungskunde) popularizes the Gauss-Jordan algorithm

kshum ENGG2013 4

http:

//en

.wik

iped

ia.o

rg/w

iki/

Wilh

elm

_Jor

dan

22 Determinant

• Area of parallelogram

kshum ENGG2013 5

(a,b)

(c,d)

Row 1 is the first vectorRow 2 is the second vector

d

b

c

a

22 Determinant

• Area of parallelogram

kshum ENGG2013 6

(3,1)

(2,4)

Row 1 is the first vectorRow 2 is the second vector

Area = 10

Outline

• Computing 22 matrix inverse via determinant

• Properties of determinant• Explain why the absolute value of determinant

is the area of parallelogram?

kshum ENGG2013 7

Determinant of 22 matrix

• Notation: Given a 22 matrix we use the notation

to stand for the determinant ps – qr.

kshum ENGG2013 8

or

Example

kshum ENGG2013 9

Determinant of identity matrix is 1

1

1

Example

kshum ENGG2013 10

Determinant of a diagonal matrixis the area of a rectangle

w

h

A formula for the Inverse of a 2x2 matrix

• Given 22 matrix

• Want to find the inverse of A

• A formula for A-1: If det A is nonzero, we have

kshum ENGG2013 11

How to compute the inverseof a 2x2 matrix

1. Exchange the two diagonal entries a, d.2. Take the negative of the two off-diagonal

entries b, c.3. Divide by the determinant.kshum ENGG2013 12

Application in solving equations

• Solve

• If we know the inverse of the 2x2 matrix, we can solve the linear system easily.

kshum ENGG2013 13

Properties of determinant

1. A matrix and its transpose has the same determinant

kshum ENGG2013 14

“The transpose of a matrix”means reflecting the matrixalong the diagonal.

Row 1 and row 2 becomecolumn 1 and column 2, andvice versa.

We write AT for the transpose of matrix A.

Proof is obvious, because

Transposing does not change area

kshum ENGG2013 15

(3,1)

(2,4)

Area = 10

(3,2)

(1,4)

Area = 10

Meta-property

Any row property of determinant is a column

property, and vice versa

kshum ENGG2013 16

By property 1, we have:

Properties of determinant

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

kshum ENGG2013 17

Zero area

Properties of determinant

3. If the two columns (or two rows) are constant multiple of each other, the determinant is zero.

kshum ENGG2013 18

Zero area

Properties of Determinant

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

kshum ENGG2013 19

(2,1)

(1,3)

The first kindof elementaryrow operation

Properties of Determinant

5. If we multiply a row (or a column) by a constant c, the value of determinant also increase by a factor of c.

kshum ENGG2013 20

The 2nd kindof elementaryrow operation

(0,1) (1,1)

(4,4)

Properties of Determinant

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

kshum ENGG2013 21

The 3rd kindof elementaryrow operation

(1,0)

(0,1) (3,1)

Properties of Determinant

7. In the linear transformation represented by a 22 matrix, the magnitude of determinant measures the area expanding factor.

kshum ENGG2013 22

Multiplied by

Square withunit area

= ad – bcThe ratio of area

(0,1) (1,1)

(1,0)

Properties of Determinant8. For any 2x2 matrices A and B, we have the

following multiplicative property

kshum ENGG2013 23

Multiplied by A

Square withunit area

Parallelogramarea = det(A)

Expand by a factor of det(A)

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

Multiplied by B

Expand by a factor of det(B)

Expand by a factor of det(AB) = det(A) det(B)

Proof of property 8

• Let

kshum ENGG2013 24

Determinant as area

• Using the properties in previous pages, we are now ready to show that the absolute value of det(M) is equal to the area of parallelogram whose sides are the two rows of M.

• We divide the argument into two steps– The two rows are perpendicular (special case).– The two rows are not perpendicular (general

case).

kshum ENGG2013 25

Determinant as area (I)• Suppose that the two rows are perpendicular i.e., ac+bd = 0 (dot product of [a b] and [c d] are zero)• Let

• Want to show that

kshum ENGG2013 26

(a,b)(c,d)

The trick is to show instead.

Determinant as area (I)

kshum ENGG2013 27

By Property 1

Just write down M and MT

By Property 8

By the definition ofmatrix multiplication

Because the dot productac+bd is zero by assumption

By the definition of determinant

By Pythagoras theorem, are the height and width of the rectangle.

Determinant as area (II)• Suppose that the two rows of M are not perpendicular.

• Idea: “Slide” the parallelogramto a rectangle, whilekeeping the areaunchanged.

kshum ENGG2013 28

(a,b)

(c,d)

Determinant as Area (II)

kshum ENGG2013 29

Decompose [c d] into two components, one is perpendicular to [a b],and the other along the same direction as [a b].

(a,b)

(c,d)

Find the constant k (hoemwork exercise)

Perpendicular to [a b](The height of parallelogram)

In the same direction as [a b]

Determinant as area (II)

• Choose a constant k such that

• Let• [c’ d’] is perpendicular to [a b] by our choice of

k.• By definition the area of parallelogram is

equal to• But

kshum ENGG2013 30

By property 6 By the first part of our proof in p.27

• We have proved the following theorem (see the picture in p.5)

Theorem: For 2x2 matrix M, the absolute value of det(M) is equal to the area of the parallelogram whose sides are the two rows (or the two columns) of M.

kshum ENGG2013 31