Chapter 2Chapter 2

DeterminantsDeterminants

With each square matrix it is possible to associate a real

number called the determinant of the matrix.

The value of this number will tell us whether the matrix

is singular.

Case 1: 1×1 Matrices If A=(a) is a 1×1 matrix, then A will have a multiplicative inverse if and only if a≠0. Thus, if we define

det(A)=a

Then A will be nonsingular if and only if det(A) ≠0.

Case 2 : 2×2 Matrices Let

then

2221

1211

aa

aaA

21122211)det( aaaaA

1 The Determinant of A Matrix

Case 3 3×3 Matrices

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

322311332112312213322113312312332211)det( aaaaaaaaaaaaaaaaaaA

Definition

Let A=(aij) be an nxn matrix and let Mij denote the (n-1)x(n-1) matrix obtained from A by deleting the row and column containing aij. The determinant of Mij is called the minor of aij. We define the cofactor Aij of aij by

( 1)i jij ijA M

Example If , then calculate det(A).

645

213

452

A

Definition The determinant of an nxn matrix A, denoted det(A), is a scalar associated with the matrix A that is defined inductively as follows:

are the cofactors associated with the entries in the first row of A.

1

1)det(

1112121111

11

nifAaAaAa

nifaA

nn

njMA jj

j ,,1)det()1( 11

1 where

Theorem 2.1.1 If A is an nxn matrix with n≥2, then det(A) can be expressed as a cofactor expansion using any row or column of A.

.,,1,,1

)(det

2211

2211

njandnifor

AaAaAa

AaAaAaA

njnjjjjj

ininiiii

Example Evaluate

3102

3010

0540

0320

Theorem 2.1.2 If A is an nxn matrix, then det(AT)=det(A).

Theorem 2.1.3 If A is an nxn triangular matrix, the determinant of A equals the product of the diagonal elements of A.

Theorem 2.1.4 Let A be an nxn matrix,(1) If A has a row or column consisting entirely of zeros, then det(A)=0.(2) If A has two identical rows or two identical columns, then det(A)=0.

2 Properties of Determinants

Lemma 2.2.1 Let A be an nxn matrix. If Ajk denotes the cofactor of ajk for k=1, … , n, then

(1)

jiif

jiifAAaAaAa jninjiji 0

)det(2211

Effects of row operation on the the value of a determinant

Row Operation I (Two rows are interchanged.)

Suppose that E is an elementary matrix of type I, then

)det()det()det()det( AEAEA

Effects of row operation on the the value of a determinant

Row Operation II (A row of A is multiplied by a nonzero constant.)

Let E denote the elementary matrix of type II formed from I by

multiplying the ith row by the nonzero constant .

)det()det()det()det( AEAEA

Row Operation III (A multiple of one row is added to another row.)

Let E be the elementary matrix of type III formed from I byadding c times the ith row to the jth row.

)det()det()det()det( AEAEA

Effects of row operation on the the value of a determinant

Ⅰ. Interchanging two rows (or columns) of a matrix

changes the sign of the determinant.

Ⅱ. Multiplying a single row or column of a matrix by a

scalar has the effect of multiplying the value of the

determinant by that scalar.

Ⅲ. Adding a multiple of one row (or column) to another

does not change the value of the determinant.

Example Evaluate

436

124

312

Main Results

Theorem 2.2.2 An n×n matrix A is singular if and only if

det(A)=0

Theorem 2.2.3 If A and B are n×n matrices, then

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

Definition The Adjoint of a MatrixLet A be an n×n matrix. We define a new matrix called theadjoint of A by

3 Cramer’s Rule

0)det()det(

11 AwhenAadjA

A

nnnn

n

n

AAA

AAA

AAA

adjA

21

22212

12111

Example Let

321

223

212

A

Compute adj A and A-1.

Theorem 2.3.1 (Cramer’s Rule) Let A be an

nxn nonsingular matrix, and let b∈Rn. Let Ai be

the matrix obtained by replacing the ith column

of A by b. If x is the unique solution to Ax=b, th

en

nifor

A

Ax ii ,,2,1

)det(

)det(

Example Use Cramer’s rule to solve

932

622

52

321

321

321

xxx

xxx

xxx