## Introduction to Linear Algebra, 2nd editionThis popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra. |

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Determinant

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Jump to: navigation, searchIn algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.

For a fixed positive integer n, there is a unique determinant function for the n×n matrices over any commutative ring R. In particular, this function exists when R is the field of real or complex numbers.

Contents

[hide]

1 Vertical bar notation

2 Determinants of 2-by-2 matrices

3 Determinants of 3-by-3 matrices

4 Applications

5 General definition and computation

6 Example

7 Properties

7.1 Useful identities

7.2 Block matrices

7.3 Relationship to trace

7.4 Derivative

8 Abstract formulation

9 Algorithmic implementation

10 History

11 See also

12 References

13 External links

[edit] Vertical bar notation

The determinant of a matrix A is also sometimes denoted by |A|. This notation can be ambiguous since it is also used for certain matrix norms and for the absolute value. However, often the matrix norm will be denoted with double vertical bars (e.g., ||A||) and may carry a subscript as well. Thus, the vertical bar notation for determinant is frequently used (e.g., Cramer's rule and minors).

For example, for matrix

the determinant det(A) might be indicated by | A | or more explicitly as

That is, the square braces around the matrices are replaced with elongated vertical bars.

[edit] Determinants of 2-by-2 matrices

The area of the parallelogram is the determinant of the matrix formed by the vectors representing the parallelogram's sides.The 2×2 matrix,

has determinant

The interpretation when the matrix has real number entries is that this gives the oriented area of the parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d). The oriented area is the same as the usual area, except that it is negative when the vertices are listed in clockwise order.

The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product xTAT, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors and in turn. These row vectors define the vertices of the unit square. With the more common matrix-vector product Ax, the parallelogram has vertices at and (note that Ax = (xTAT)T).

A formula for larger matrices will be given below.

[edit] Determinants of 3-by-3 matrices

The volume of this Parallelepiped is the absolute value of the determinant of the matrix formed by the rows r1, r2, and r3.The 3×3 matrix:

Using the cofactor expansion on the first row of the matrix we get:

The determinant of a 3x3 matrix can be calculated by its diagonals.which can be remembered as the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements when the copies of the first two columns of the matrix are written beside it as below:

Note that this mnemonic does not carry over into higher dimensions.

[edit] Applications

Determinants are used to characterize invertible matrices (i.e., exactly those matrices with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial

where I is the identity matrix of the same dimension as A.

One often thinks of the determinant as assigning a number to every sequence of n vectors in , by using the square matrix whose columns

### Contents

Chapter One A SYSTEM OF VECTORS 1 Introduction | 1 |

Description of the system 3 | 2 |

Directed line segments and position vectors | 3 |

Addition and subtraction of vectors | 5 |

Multiplication of a vector by a scalar | 8 |

Section formula and collinear points | 10 |

Centroids of a triangle and a tetrahedron | 12 |

Coordinates and components | 14 |

Direct sums of subspaces | 126 |

Exercises on chapter 5 | 130 |

Chapter Six LINEAR MAPPINGS 44 Introduction | 133 |

Some examples of linear mappings | 134 |

Some elementary facts about linear mappings | 136 |

New linear mappings from old | 138 |

Image space and kernel of a linear mapping | 140 |

Rank and nullity | 144 |

Scalar products | 18 |

Postscript | 22 |

Exercises on chapter 1 | 24 |

Chapter Two MATRICES 11 Introduction | 26 |

Basic nomenclature for matrices | 27 |

Addition and subtraction of matrices | 29 |

Multiplication of a matrix by a scalar | 30 |

Multiplication of matrices | 31 |

Properties and nonproperties of matrix multiplication | 33 |

Some special matrices and types of matrices | 37 |

Transpose of a matrix | 39 |

First considerations of matrix inverses | 41 |

Properties of nonsingular matrices | 43 |

Partitioned matrices | 45 |

Exercises on chapter 2 | 49 |

Chapter Three ELEMENTARY ROW OPERATIONS 22 Introduction | 52 |

Some generalities concerning elementary row operations | 53 |

Echelon matrices and reduced echelon matrices | 55 |

Elementary matrices | 61 |

Major new insights on matrix inverses | 63 |

Generalities about systems of linear equations | 67 |

Elementary row operations and systems of linear equations | 70 |

Exercises on chapter 3 | 77 |

Chapter Four AN INTRODUCTION TO DETERMINANTS 29 Preface to the chapter | 79 |

Minors cofactors and larger determinants | 80 |

Basic properties of determinants | 83 |

The multiplicative property of determinants | 87 |

Another method for inverting a nonsingular matrix | 90 |

Exercises on chapter 4 | 92 |

Chapter Five VECTOR SPACES 34 Introduction | 94 |

The definition of a vector space and examples | 95 |

Elementary consequences of the vector space axioms | 97 |

Subspaces | 99 |

Spanning sequences | 102 |

Linear dependence and independence | 106 |

Bases and dimension | 112 |

Further theorems about bases and dimension | 118 |

Sums of subspaces | 122 |

Rowand columnrank of a matrix | 147 |

Systems of linear equations revisited | 150 |

Rank inequalities | 154 |

Vector spaces of linear mappings | 158 |

Exercises on chapter 6 | 161 |

Chapter Seven MATRICES FROM LINEAR MAPPINGS 54 Introduction | 164 |

The main definition and its immediate consequences | 165 |

Matrices of sums etc of linear mappings | 170 |

Change of basis | 172 |

Matrix of a linear mapping w r t different bases | 174 |

The simplest matrix representing a linear mapping | 176 |

Vector space isomorphisms | 178 |

Exercises on chapter 7 | 181 |

Chapter Eight EIGENVALUES EIGENVECTORS AND DIAGONALIZATION 61 Introduction | 184 |

Eigenvalues and eigenvectors | 187 |

Eigenvalues in the case F C | 193 |

Diagonalization of linear transformations | 196 |

Diagonalization of square matrices | 199 |

The hermitian conjugate of a complex matrix | 201 |

Eigenvalues of special types of matrices | 202 |

Exercises on chapter 8 | 204 |

Chapter Nine EUCLIDEAN SPACES 69 Introduction | 207 |

Some elementary results about euclidean spaces | 209 |

Orthonormal sequences and bases | 211 |

Lengthpreserving transformations of a euclidean space | 215 |

Orthogonal diagonalization of a real symmetric matrix | 219 |

Exercises on chapter 9 | 224 |

Chapter Ten QUADRATIC FORMS 74 Introduction | 226 |

Change of basis and change of variable | 230 |

Diagonalization of a quadratic form | 233 |

Invariants of a quadratic form | 239 |

Orthogonal diagonalization of a real quadratic form | 242 |

Positivedefinite real quadratic forms | 245 |

The leading minors theorem | 249 |

vii | 254 |

Appendix MAPPINGS | 257 |

273 | |