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Chapter 6 – Determinant

Outline6.1 Introduction to Determinants6.2 Properties of the Determinant6.3 Geometrical Interpretations of the Determinant; Cramer’s Rule

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6.1 Introduction to Determinants

• A matrix is invertible if (and only if) . The quantity ad-bc is called the determinants of the matrix A, denoted by det(A).

• If the matrix A is invertible, then its inverse can be expressed in terms of the determinant:

• Sarrus’s rule– To find the determinant of a 3 × 3 matrix A, write the first two columns

of A to the right of A. Then, multiply the entries along the six diagonals:

– Add or subtract these diagonal products as shown in the diagram.

– det(A) = a11a22a33+a12a23a31+a13a21a32-a13a22a31-a11a23a32-a12a21a33

dc

baA 0 bcad

ac

bd

Aac

bd

bcadA

)det(

111

3

Example 3

• For which choices of the scalar λ is the matrix invertible

• (sol)– det(A) = (1-λ)3+1+1-(1-λ)-(1-λ)-(1-λ)

=1-3λ+3λ2-λ3+2-3+3λ=-λ3+3λ2

=λ2(3-λ).

– The determinant is 0 if λ = 0 or λ = 3. The matrix A is invertible if λ is neither 0 nor 3.

111

111

111

A

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The Determinant of an n×n Matrix

• A pattern in an n × n matrix is a way to choose n entries of the matrix so that there is one chosen entry in each row and one in each column of the matrix. Two numbers in a pattern are inverted if one of them is located to the right and above the other in the matrix. We obtain the determinant of an n × n matrix A by summing the products associated with all patterns with an even number of inversions and subtracting the products associated with all patterns with an odd number of inversions.

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Example 4

• Count the number of inversions in the following pattern:

• (sol)– There are seven inversions, as indicated by the bars:

432123

456789

876543

212345

678987

654321

432123

456789

876543

212345

678987

654321

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Example 7

• Find det(A) for

• (sol)– only one pattern makes a nonzero contribution toward the determinant:

– Thus, det(A)=-3·2·1·8·5·2 = -480.

000100

050000

000003

200000

008000

000020

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Example 8

• Find det(A) for

• (sol)– Again, only one pattern makes a nonzero contribution. In the second

column, we must choose the second component, 3. Then, in the fourth column, we must choose the third component, 2. Next, think about the last column; and so on:

– det(A)=-6·3·4·2·1=-144.

10505

00400

92308

73239

00106

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Example 9

• Find det(A) for

• (sol)– Note that A is an upper triangular matrix. To make a nonzero

contribution, a pattern must contain the first component of the first column, then the second component of the second column, and so on. Thus, only the diagonal pattern makes a nonzero contribution. We conclude that

– det(A)=product of the diagonal entries=1·2·3·4·5=120.

• The determinant of an (upper or lower) triangular matrix is the product of the diagonal entries of the matrix.

50000

54000

54300

54320

54321

A

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Example 10

• Find det(A) for

• (sol)– Most of the 4! = 24 patterns in this matrix contain zeros. To get a pattern

that could be nonzero, we have to choose the entry n in the last row and the k in the third row. In the first two rows, we are then left with two choices:

– Thus, det(A)=afkn-ebkn=(af-eb)kn. Note that the first factor is the

– determinant of the matrix

n

mk

hgfe

dcba

A

000

00

fe

ba

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6.2 Properties of the Determinant

• (Determinant of the transpose) If A is a square matrix, then det(AT)=det(A).

• The function T(A)=det(A) from Rn×n to R is nonlinear (if n > 1).

• (Linearity of the determinant in the columns)

• Consider the vectors , in Rn. Then, the transformation

from Rn to R is linear. This property is called linearity of the determinant in the jth column. Likewise, the determinant is linear in the rows.

njj vvvv

,,,,, 111

|||||

|||||

det)( 111 njj vvxvvxT

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Example 2

• Consider the transformation from R4 to R. Is this transformation linear?

• (sol)– Each pattern in the matrix

involves three numbers and one of the variables xi ; that is, the product associated with a pattern has the form kxi , for some constant k. The diagonal pattern, for example, makes the contribution 5x3. The determinant itself, obtained by adding and subtracting such products, has the form det(A)=c1x1+c2x2+c3x3+c4x4, for some constants ci . Therefore, the transformation T is indeed linear.

134

567

654

321

det

4

3

2

1

4

3

2

1

x

x

x

x

x

x

x

x

T

134

567

654

321

4

3

2

1

x

x

x

x

A

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Linearity of the determinant in the columns

• If three n×n matrices A, B, C are the same, except for the jth column, and the jth column of C is the sum of the jth columns of A and B, then det(C)=det(A)+det(B).

• If two n×n matrices A and B are the same, except for the jth column, and the j th column of B is k times the jth column of A, then det(B)=kdet(A):

||

|||

det

||

|||

det

||

|||

det 111 nnn vyvvxvvyxv

||

|||

det

||

|||

det 11 nn vxvkvxkv

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Determinants and Gauss–Jordan Elimination

• (Elementary row operations and determinants)– If B is obtained from A by dividing a row of A by a scalar k, then

det(B)=(1/k)det(A).

– If B is obtained from A by a row swap, then det(B)=-det(A).

– If B is obtained from A by adding a multiple of a row of A to another row, then det(B)=det(A).

– Analogous results hold for elementary column operations.

• A square matrix A is invertible if and only if det(A)≠0.• Consider an invertible matrix A. Suppose you swap

rows s times as you row-reduce A and you divide various rows by the scalars k1, k2, . . . , kr . Then, det(A)=(-1)sk1k2...kr .

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Example 5• Find

• (sol)– We perform Gauss–Jordan elimination, keeping a note of all row swaps

and row divisions we do:–

– We made two swaps and performed row divisions by 2, -1, and -2, so that

– det(A)=(-1)2·2·(-1)·(-2)=4.

2111

1211

1211

6420

det

2000

1100

3210

1211

)2(

)1(

2000

1100

3210

1211

1100

2000

3210

1211

)(

)(

2111

1211

3210

1211

2

2111

1211

6420

1211

2111

1211

1211

6420

I

I

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Determinants

• (Determinant of a product)– If A and B are n×n matrices, then det(AB)=det(A)det(B).

• (Determinant of an inverse)– If A is an invertible matrix, then

• (Example 6) If matrix A is similar to B, what is the relationship between det(A) and det(B)?

• (sol)– There is an invertible matrix S such that B = S-1AS. Now,

– det(B)=det(S-1AS)=det(S-1)det(A)det(S)= (det S)-1det(A)det(S)= det(A). (The determinants commute, because they are scalars.)

– Thus, det(B)=det(A).

)det(

11)(det)det( 1

AAA

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Minors

• Recall the formula– det(A)=a11a22a33+a12a23a31+a13a21a32-a13a22a31-a11a23a32-a12a21a33

– det(A)=a11(a22a33-a32a23)+a21(a32a13-a12a33)+a31(a12a23-a22a13).

•

• (Minors) For an n×n matrix A, let Aij be the matrix obtained by omitting the ith row and the jth column of A. The (n-1)×(n-1) matrix Aij is called a minor of A.

• det(A)=a11det(A11)-a21det(A21)+a31det(A31).

• This representation of the determinant is called the Laplace expansion of det(A) down the first column.

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Example 7

• Consider an n × n matrix A whose j th column is :

What is the relationship between det(A) and det(Aij)?

ie

nnnn

inii

n

n

aaa

aaa

aaa

aaa

0

1

0

0

21

21

22221

11211

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Example 7 (II)

• (sol)– As we compute det(A), we need to consider only those patterns in A that involve

the 1 in the jth column, since all other patterns will contain a zero. Each such pattern corresponds to a pattern of Aij , involving the same numbers, except for the 1 we have omitted. For example,

–

– The products associated with the two patterns are the same, but what happens to the number of inversions? We lose all the inversions involving the 1 in the jth column. The number of such inversions is even if i + j is even and odd if i + j is odd. (In our example, i + j = 7 is odd, and we are losing 3 inversions.) Since these remarks apply to all the patterns of A involving the 1 in the jth column, we can conclude that det(A)=±det(Aij), taking the plus sign if i+j is even and the negative sign if i+j is odd. We can write this more succinctly as )det()1()det( ij

ji AA

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Laplace Expansion

• We can compute the determinant of an n×n matrix A by Laplace expansion along any row or down any column.

• Expansion along the ith row:

• Expansion down the j th column:

• The signs follow a checkerboard pattern:

n

jijij

ji AaA1

)det()1()det(

n

iijij

ji AaA1

)det()1()det(

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Example 8

• Use Laplace expansion to compute det(A) for

• (sol)–

3005

0229

0319

2101

A

55)1830(22120

)39

11det3

05

39det2(2

29

11det3

05

29det2

305

039

211

det2

305

029

211

det

3005

0229

0319

2101

det2

3005

0229

0319

2101

det1

)det()det()det()det()det( 4242323222221212

AaAaAaAaA

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The Determinants of a Linear Transformation

• Consider a linear transformation T from V to V, where V is a finite-dimensional linear space. If B is a basis of V and B is the B-matrix of T , then we definedet(T)=det(B).This determinant is independent of the basis B we choose.

• (Example 9) Let V be the space spanned by functions cos(2x) and sin(2x). Find the determinant of the linear transformation D(f)=f ‘ from V to V.

• (sol)– The matrix B of D with respect to the basis cos(2x), sin(2x) is

– so that det(D)=det(B)=4.

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20B

226.3 Geometrical Interpretations the Determinant;Cramer’s Rule

• The determinant of an orthogonal matrix is either 1 or -1.

• An orthogonal n×n matrix A with det(A)=1 is called a rotation matrix, and the linear transformation is called a rotation.

• Consider a 2×2 matrix . Then, the area of the parallelogram defined by and is |det(A)|.

• We see that if the direction of is obtained by rotating through a positive (i.e., counterclockwise) angle between 0 and π, then will be positive. If we rotate through a negative (clockwise) angle between 0 and .π, then det(A) will be negative.

xAxT

)(

21 vvA

1v

2v

1v

1v

21det)det( vvA

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The Area of the Parallelogram

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The Area of the Parallelogram (II)

• If A is an n×n matrix with columns , then ,where .

• Consider a 3×3 matrix . Then, the volume of the parallelepiped defined by , , and is |det(A)|.

nvv

,,1

nVnV vvvvvAn

11

projproj)det( 221

),,,(span 21 jj vvvV

321 vvvA

1v

2v

3v

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k-parallelepipeds and k-volume

• Consider the vectors in Rn. The k-parallelepiped defined by the vectors is the set of all vectors in Rn of the form , where 0 ≤ ci ≤ 1. The k-volume of this k-parallelepiped is defined recursively by and

• Consider the vectors in Rn. Then the k-volume of the k-parallelepiped defined by the vectors is ,where A is the n×k matrix with columns . In particular, if k=n, this volume is |det(A)|.

kvvv

,,, 21

kvvv

,,, 21

kkvcvcvc

2211

),,,( 21 kvvvV

11)( vvV

kVkkk vvvvvVvvvV

k

1

proj),,,(),,,( 12121

kvvv

,,, 21

kvvv

,,, 21

)det( AAT

kvvv

,,, 21

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The Determinant as Expansion Factor

• Consider a linear transformation from R2 to R2. Then, |det(A)| is the expansion factor

of T on parallelograms Ω.Likewise, for a linear transformation from Rn to Rn, |det(A)| is the expansion factor of T on n-parallelepipeds:

for all vectors in Rn.

xAxT

)(

of area

)( of area T

xAxT

)(

),,()det(),,( 11 nn vvVAvAvAV

nvv

,,1

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Cramer’s Rule

• Consider the linear systemwhere A is an invertible n×n matrix. The components xi of the solution vector are

where Ai is the matrix obtained by replacing the ith column of A by .

• (Corollary to Cramer’s rule) Consider an invertible n×n matrix A. The classical adjoint adj(A) is the n×n matrix whose ijth entry is (-1)i+j det(Aji). Then,

bxA

x

b

)det(

)det(

A

Ax i

i

)(adj)det(

11 AA

A

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Example 3

• Use the preceding formula to solve the system

• (sol)–

1354

732

21

21

xx

xx

,2

54

32det

513

37det

1

x 1

54

32det

134

72det

2

x

29

Example 4

• Solve the systemwhere a, b, C are arbitrary positive constants.

• (sol)–

,)1(

0)1(

21

21

Cxbax

axxb

221 )1(

1

1det

1

0det

ab

aC

ba

ab

bC

a

x

222 )1(

)1(

1

1det

01det

ab

Cb

ba

ab

Ca

b

x

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Example 5

• Consider the linear system

How does the solution x change as we change the parameters a and c? More precisely, find ∂x/∂a and ∂x/∂c, and determine the signs of these quantities.

• (sol)–

.0 and 0 where,1

1

cabddycx

byax

,0

det

1

1det

bcad

bd

dc

ba

d

b

x

0)(

)(2

bcad

bdd

a

x0

)(

)(2

bcad

bdb

c

x

31

Example 5 (II)

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Example 6

• For the vectors , and shown in Figure 10, consider the linear system , where .Using the terminology introduced in Cramer’s rule, let Note that det(A) and det(A2) are both positive, according to the criteria we discussed after Fact 6.3.3. Cramer’s rule tells us that

Explain this last equation geometrically, in terms of areas of parallelograms.

• (sol)– We can write the system as

– Now,

21, ww

b

bxA

21 wwA

bwA

12

).det()det(or )det(

)det(22

22 AxA

A

Ax

bxA

bwxwx

2211

),det() and by defined ramparallelog theof (area

and by defined ramparallelog theof area

and by defined ramparallelog theof areadet)det(

2212

221

112

Axwwx

wxw

bwbwA

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Example 6 (II)