Chapter 2 Determinants

DETERMINANTS BY COFACTOR EXPANSION 2.1 DETERMINANTS BY COFACTOR EXPANSION For A (2x2) matrix The expression is called the determinant of the matrix and is denoted by the symbol det A or |A| the formula for A-1 given in Theorem 1.4.5 is

Minors and Cofactors

EXAMPLE 1 Finding Minors and Cofactors Let

Cofactor Expansions The definition of a 3x3 determinant in terms of minors and cofactors is the determinant of an nxn matrix to be This method of evaluating det (A) is called cofactor expansion along the first row of A.

EXAMPLE 2 Cofactor Expansion Along the First Row Let . Evaluate det (A) by cofactor expansion along the first row of A.

All of the following are correct for 3x3 A

EXAMPLE 3 Cofactor Expansion Along the First Column Let A be the matrix in Example 2. Evaluate det (A) by cofactor expansion along the first column of .

EXAMPLE 4 Smart Choice of Row or Column If is the 4X4 matrix then to find det(A) it will be easiest to use cofactor expansion along the second column, since it has the most zeros: For the determinant, it will be easiest to use cofactor expansion along its second column, since it has the most zeros:

Adjoint of a Matrix

EXAMPLE 6 Adjoint of a Matrix Let The cofactors of A are so the matrix of cofactors is and the adjoint of A is

THEOREM 2.1.2 Inverse of a Matrix Using Its Adjoint If A is an invertible matrix, then

EXAMPLE 7 Using the Adjoint to Find an Inverse Matrix Use 7 to find the inverse of the matrix A in Example 6.

THEOREM 2.1.3

EXAMPLE 8 Determinant of an Upper Triangular Matrix

Cramer's Rule

EXAMPLE 9 Using Cramer's Rule to Solve a Linear System Use Cramer's rule to solve

Let A be a square matrix. Then 2.2 EVALUATING DETERMINANTS BY ROW REDUCTION THEOREM 2.2.1 THEOREM 2.2.2 Let A be a square matrix. Then

Elementary Row Operations

THEOREM 2.2.4

EXAMPLE 2 Determinants of Elementary Matrices The following determinants of elementary matrices, which are evaluated by inspection, illustrate Theorem 2.2.4.

Matrices with Proportional Rows or Columns THEOREM 2.2.5 If A is a square matrix with two proportional rows or two proportional columns, then EXAMPLE 3 Introducing Zero Rows The following computation illustrates the introduction of a row of zeros when there are two proportional rows: Each of the following matrices has two proportional rows or columns; thus, each has a determinant of zero.

Evaluating Determinants by Row Reduction EXAMPLE 4 Using Row Reduction to Evaluate a Determinant Evaluate det(A) where

Reduced A to row-echelon form (which is upper triangular) and apply Theorem 2.2.3:

EXAMPLE 5 Using Column Operations to Evaluate a Determinant Compute the determinant of

EXAMPLE 6 Row Operations and Cofactor Expansion Evaluate det (A) where

2.3 PROPERTIES OF THE DETERMINANT FUNCTION Basic Properties of Determinants For example,

EXAMPLE 1 Consider

THEOREM 2.3.1 EXAMPLE 2 Using Theorem 2.3.1

THEOREM 2.3.3 A square matrix A is invertible if and only if

EXAMPLE 3 Determinant Test for Invertibility Since the first and third rows of are proportional, . Thus A is not invertible.

THEOREM 2.3.4

THEOREM 2.3.5 If A is invertible, then

Linear Systems of the Form Many applications of linear algebra are concerned with systems of n linear equations in n unknowns that are expressed in the form where λ is a scalar EXAMPLE 5 Finding The linear system can be written in matrix form as

This is called the characteristic equation of A EXAMPLE 6 Eigenvalues and Eigenvectors Find the eigenvalues and corresponding eigenvectors of the matrix A in Example 5.

The eigenvectors of A corresponding to λ=-2 are

THEOREM 2.3.6

2.4 A COMBINATORIAL APPROACH TO DETERMINANTS EXAMPLE 7 Determinants of 2x2 and 3x3 Matrices Warning the methods do not work for determinants of 4x4 matrices or higher.

EXAMPLE 8 Evaluating Determinants

C H A P T E R 3 Vectors in 2-Space and 3-Space

3.1 INTRODUCTION TO VECTORS (GEOMETRIC)

DEFINITION If v and w are any two vectors, then the sum v + w is the vector determined as follows: Position the vector w so that its initial point coincides with the terminal point of v. The vector v + w is represented by the arrow from the initial point of v to the terminal point of w

Vectors in Coordinate Systems

Vectors in 3-Space each point P in 3-space has a triple of numbers (x, y, z), called the coordinates of P

In Figure a the point (4, 5, 6) and in Figure b the point (-3 , 2, -4).

EXAMPLE 1 Vector Computations with Components

EXAMPLE 2 Finding the Components of a Vector

Translation of Axes The solutions to many problems can be simplified by translating the coordinate axes to obtain new axes parallel to the original ones. These formulas are called the translation equations.

EXAMPLE 3 Using the Translation Equations

3.2 NORM OF A VECTOR; VECTOR ARITHMETIC Properties of Vector Operations THEOREM 3.2.1 Properties of Vector Arithmetic If u, v, and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold.

Norm of a Vector

EXAMPLE 1 Finding Norm and Distance