Chapter 2Determinants

The Determinant Function– The 22 matrix

is invertible if ad-bc0. The expression ad-bc occurs so frequently that it has a name; it is called the determinant of the matrix A and is denoted by the symbol det(A).

–We want to obtain the analogous formula to square matrices of higher order.

dc

baA

Cofactor Expansion: Cramer's Rule

• Minors and Cofactors– det(A)=a11a22a33+a12a23a31+a13a21a32-a13a22a31-a12a21a33-a11a23a32 =a11(a22a33-a23a32)-a12(a21a33-a23a31)+a13(a21a32-a22a31)

=a11M11-a12M12+a13M13 where

– Definition: If A is a square matrix, then the minor of entry aij is denoted by Mij and is defined to be the determinant of the submatrix that remains after the ith row and jth column are deleted from A. The number (-1)i+jMij is denoted by Cij and is called the cofactor of entry aij.

• Example: Finding Minors and Cofactors

3231

222113

3331

232112

3332

232211 , ,

aa

aaM

aa

aaM

aa

aaM

841

652

413

A

– The cofactor and minor of an element differ only in sign. To determine the sign relating Cij and Mij, we may use the checkerboard array

• Cofactor Expansions– det(A)=a11M11-a12M12+a13M13=a11C11+a12C12+a13C13

– This method of evaluating det(A) is called cofactor expansion along the first row of A.• Example: Evaluate det(A) by cofactor expansion

along the first row of A.

Cofactor Expansion: Cramer's Rule

245

342

013

A

– Theorem: The determinant of an nn matrix A can be computed by multiplying the entries in any row(or column) by their cofactors and adding the resulting products; that is, for each iin and 1jn,

• Example: Cofactor expansion along the first column– Cofactor expansion and row or column operations can

sometimes be used in combination to provide an effective method for evaluating determinant.• Example: Evaluate det(A) where

Cofactor Expansion: Cramer's Rule

ininiiii

njnjjjjj

CaCaCaA

CaCaCaA

2211

2211

)det(

)det(

3573

5142

1121

6253

A

• Adjoint of a Matrix– If we multiply the entries in any row by the corresponding

cofactors from a different row, the sum of these products is always zero. (This result also holds for columns)• Example: Find a11C31+a12C32+a13C33 for

– Definition: If A is an nn matrix and Cij is the cofactor of aij, then the matrix

is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A is denoted by adj(A).

Cofactor Expansion: Cramer's Rule

333231

232221

131211

aaa

aaa

aaa

A

nnnn

n

n

CCC

CCC

CCC

21

22221

11211

– Example: Find the adjoint of A where

– Theorem: If A is an invertible matrix, then

• Example: Find the inverse of A in last example.

• Cramer's Rule– be of marginal interest for computational

purposes, but is useful for studying the mathematical properties of a solution without the need for solving the system

Cofactor Expansion: Cramer's Rule

042

361

123

A

)adj()det(

11 AA

A

– Theorem: If Ax=b is a system of n linear equations in n unknowns such that det(A)0, then the system has a unique solution. This solutions is

where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix

Example: Use Cramer's rule to solve

Cofactor Expansion: Cramer's Rule

AA

xA

Ax

A

Ax n

n det

det,,

det

det ,

det

det 22

11

nb

b

b

2

1

b

832

30643

62

321

321

31

xxx

xxx

xx

Cramer’s Rule• Solution

• Therefore,

321

643

201

A

328

6430

206

1A

381

6303

201

2A

821

3043

601

1A

11

10

44

40

)det(

)det( 11

A

Ax

11

18

44

72

)det(

)det( 22

A

Ax

11

38

44

152

)det(

)det( 33

A

Ax

Evaluating Determinants by Row Reduction• A Basic Theorem

– Theorem: Let A be a square matrix.

(a) If A has a row of zeros or a column of zeros, then det(A)=0.

(b) det(A) = det(AT).• It is evident from (b) that every theorem about

determinants that contains the word “row” is also true when the word “column” is substituted.

• Triangular Matrices– Theorem: If A is an nn triangular matrix(upper triangular,

lower triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; that is, det(A)=a11a22...ann.

• Example: Find the determinant of

40000

89000

67600

15730

38372

• Elementary Row Operations– Theorem: Let A be an nn matrix.

(a) If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, then det(B)=k det(A).

(b) If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = -det(A).

(c) If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column is added to another column, then det(B) = det(A).

• Elementary Matrices– Theorem: Let E be an nn elementary matrix.

(a) If E results from multiplying a row of In by k, then det(E) = k.

(b) If E results from interchanging two rows of In, then det(E) = -1.

(c) If E results from adding a multiple of one row of In to another, then det(E)=1.

Evaluating Determinants by Row Reduction

• Matrices with Proportional Rows or Columns– Theorem: If A is a square matrix with two

proportional rows or two proportional columns, then det(A) = 0.

• Evaluating Determinants by Row Reduction– Idea: to reduce the given matrix to upper

triangular form by elementary row operations, then compute the determinant of the upper triangular matrix, then relate that determinant to that of the original matrix

– Example: Evaluate det(A) where

– The method is well suited for computer evaluation since it is systematic and easily programmed.

Evaluating Determinants by Row Reduction

162

963

510

A

Properties of the Determinant Function• Basic Properties of Determinants

– Let A and B be nn matrices and k be any scalar. We have det(kA)=kndet(A).

– det(A+B) is usually not equal to det(A)+det(B).

• Example:

• Interesting example:

– Theorem: Let A, B, and C be nn matrices that differ only in single row, say the rth, and assume that the rth row of C can be obtained by adding corresponding entries in the rth rows of A and B. Then det(C) = det(A) + det(B). The same result holds for columns.

31

13 ,

52

21BA

2221

1211

2221

1211 and bb

aaB

aa

aaA

• Determinant of a Matrix Product– If A and B are square matrices of the same size, then

det(AB)=det(A)det(B)– Lemma: If B is an nn matrix and E is an nn elementary

matrix, then det(EB)=det(E)det(B).

• Det(E1E2...ErB)=det(E1)det(E2)...det(Er)det(B)

• Determinant Test for Invertibility– Theorem: A square matrix A is invertible if and only if

det(A)0.– A square matrix with two proportional rows or columns

is not invertible.– Theorem: If A and B are square matrices of the same

size, then det(AB)=det(A)det(B).

Properties of the Determinant Function

– Example: Verifying det(AB)=det(A)det(B)

– Theorem: If A is invertible, then

Properties of the Determinant Function

85

31 ,

12

13BA

)det(

1)det( 1

AA

• Summary– Theorem: If A is an nn matrix, then the

following are equivalent.(a)A is invertible.

(b)Ax=0 has only the trivial solution.

(c)The reduced row-echelon form of A is In.

(d)A is expressible as a product of elementary matrices.

(e)Ax=b is consistent for every n1 matrix b.

(f) Ax=b has exactly one solution for every n1 matrix b.

(g)det(A)0.

Properties of the Determinant Function

• Definition of a Determinant– By an elementary product from an nn

matrix A we shall mean any product of n entries from A, no two of which come from the same row or same column.

The Determinant Function

• Example: List all elementary products from the matrices

– An nn matrix A has n! elementary products of the form

– signed elementary product from A: an elementary product multiplied by +1 or –1. We use + if (j1,j2,...,jn) is an even permutation and the – if (j1,j2,...,jn) is an odd permutation.

• Example: List all signed elementary products from the matrices

The Determinant Function

333231

232221

131211

2221

1211 )( )(

aaa

aaa

aaa

baa

aaa

nnjjj aaa 21 21

333231

232221

131211

2221

1211 )( )(

aaa

aaa

aaa

baa

aaa

• Definition: Let A be a square matrix. The determinant function is denoted by det, and we define det(A) to be the sum of all signed elementary products from A. The number det(A) is called the determinant of A.

• Example: Find the determinants of matrices

– mnemonic method(only for 22 and 33 matrices)

• Example: Evaluate the determinants of

The Determinant Function

333231

232221

131211

2221

1211 )( )(

aaa

aaa

aaa

baa

aaa

3231

2221

1211

333231

232221

131211

2221

1211

aa

aa

aa

aaa

aaa

aaa

aa

aa

987

654

321

and 24

13BA

• Notation and Terminology– The symbol |A| is an alternative notation for

det(A).– Although the determinant of a matrix is a

number, it is common to use the term “determinant” to refer to the matrix whose determinant is being computed.

– The determinant of A is often written symbolically as

– Evaluating determinants directly from the definition leads to computational difficulties.

The Determinant Function

nnjjj aaaA

21 21)det(