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Chapter 2 Determinants

Chapter 2 Determinants. 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.

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Chapter 2 Determinants

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  1. Chapter 2 Determinants

  2. 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

  3. Minors and Cofactors

  4. EXAMPLE 1 Finding Minors and Cofactors Let

  5. 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.

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

  7. All of the following are correct for 3x3 A

  8. 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 .

  9. 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:

  10. Adjoint of a Matrix

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

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

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

  14. THEOREM 2.1.3

  15. EXAMPLE 8 Determinant of an Upper Triangular Matrix

  16. Cramer's Rule

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

  18. 2.2 EVALUATING DETERMINANTS BY ROW REDUCTION THEOREM 2.2.1 THEOREM 2.2.2 Let A be a square matrix. Then

  19. Elementary Row Operations

  20. THEOREM 2.2.4

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

  22. 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.

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

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

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

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

  27. 2.3 PROPERTIES OF THE DETERMINANT FUNCTION Basic Properties of Determinants For example,

  28. EXAMPLE 1 Consider

  29. THEOREM 2.3.1 EXAMPLE 2 Using Theorem 2.3.1

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

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

  32. THEOREM 2.3.4

  33. THEOREM 2.3.5 If A is invertible, then

  34. 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

  35. 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.

  36. The eigenvectors of A corresponding to λ=-2 are The eigenvectors of A corresponding to λ=5 are

  37. THEOREM 2.3.6

  38. 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.

  39. EXAMPLE 8 Evaluating Determinants

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

  41. 3.1 INTRODUCTION TO VECTORS (GEOMETRIC)

  42. 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

  43. Vectors in Coordinate Systems

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