ENGG2013 Unit 9 3x3 Determinant

Feb, 2011.

Last time • • • 2  2 determinant Compute the area of a parallelogram by determinant A formula for 2x2 matrix inverse kshum ENGG2013 2

Today • • 3  3 determinant and its properties Using determinant, we can – – – test whether three vectors lie on the same plane solve 3  3 linear system test whether the inverse of a 3  3 matrix exists kshum ENGG2013 3

Vector Notation • We will use two different notations for a point in the 3D space (x,y,z) z z y x y x kshum ENGG2013 4

2  2 determinant Notation for 2  2 determinant : kshum How to calculate: ad – bc + – ENGG2013 - bc ad 5

3  3 determinant Notation for 3  3 determinant : Definition : kshum ENGG2013 6

+ Rule of Sarrus – + + – – kshum ENGG2013 Pierre Frédéric Sarrus (1798 – 1861) 7

Volume of parallelepiped • Geometric meaning – The magnitude of 3  3 determinant is the volume of a parallelepiped z y x 8 kshum ENGG2013

Co-planar  zero determinant • Determinant = 0  Volume = 0  the three vectors lie on the same plane z y x A collection of vectors are said to be

co-planar

if they lie on the same plane.

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Det of Diagonal matrix • Volume of a rectangular box c b kshum a ENGG2013 10

Transpose has the same determinant – + + + – – Compare with kshum ENGG2013 11

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 Volume of parallelepiped with vertices (0,0,0), (1,2,0) , (2,0,1) , ( –1, 1, 3) equals to the absolute value of z or x ENGG2013 kshum y 12

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 3  3 matrix

A

system

A

x =

b

by Cramer’s rule.

is non-zero, we can solve the linear The solution to or equivalently is

A x b

ENGG2013 Gabriel Cramer (1704-1752) 14 kshum

kshum

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 kshum We have already verified this property in p.11

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Meta-property • • Because 1. 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 kshum ENGG2013 19

Properties of determinant 3. 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 kshum ENGG2013 20

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 The first kind of elementary row operation 21 kshum ENGG2013

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

.

The 2nd kind of elementary row operation For example, if we multiply the third row by a constant

k

, kshum ENGG2013 22

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 For example, if the first column is the sum of two column vectors, then we have kshum ENGG2013 23

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

The 3rd kind of elementary row operation For example, if we replace the 3 rd sum of the 3 rd column and

k

column by the times the 2 nd column, kshum ENGG2013 24

Summary on the effect of the elementary row (or column) operations on determinant • • • Exchange two rows (or columns)  the sign of determinant change 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 kshum ENGG2013 25

Proof of the Cramer’s rule • The solution to is Verification for x 1 : Substitute the value of b 1 , b 2 and b 3 in the first column of A.

Verification for x 2 : Substitute the value of b 1 , b 2 and b 3 in the second column of A.

Etc.

kshum ENGG2013 Cramer’s rule in wikipedia 26

Because x 1 , x 2 , x 3 satisfy the system of linear equations, we have kshum =0 ENGG2013 By substitution Property 6 Property 5 =0 By Property 3 27

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MINOR AND COFACTOR

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Another way to compute det Group the six terms as kshum 3  3 determinant can be computed in terms of 2  2 determinant ENGG2013 29

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 A ij . It is also called the minor of the (i,j)-entry a ij in A. kshum ENGG2013 30

Expansion on the first row kshum Minor of a 1 Minor of b 1 Minor of c 1 ENGG2013 31

Expansion on the second row kshum Minor of a 2 Minor of b 2 Minor of c 2 ENGG2013 32

Expansion on the third row kshum Minor of a 3 Minor of b 3 Minor of c 3 ENGG2013 33

kshum The sign pattern ENGG2013 Expansion on the first row Expansion on the second row Expansion on the last row 34

Column expansion We have similar recursive formula for determinant by column expansion For example, kshum Computation on the third column is easy, because there are lots of zeros.

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Cofactor • • The minor together with the appropriate  cofactor .

sign is called For The sign The cofactor of C ij is defined as Expansion on the i-th row (i=1,2,3): The minor of a ij Expansion on the j-th column (j=1,2,3): kshum ENGG2013 36

A formula for matrix inverse Suppose that det

A

is nonzero.

Usually called the adjoint of

A

Three steps in computing above formula 1. for i,j = 1,2,3, replace each a ij by cofactor C ij 2. Take the transpose of the resulting matrix.

3. divide by the determinant of

A

.

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A Quotation Algebra is but written geometry; geometry is but drawn algebra.

--- Sophie Germain (1776-1831) L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée kshum ENGG2013 38