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Chapter 3

Chapter 3. Determinants Gareth Williams J & B, 滄海書局. 3.1 Introduction to Determinants 3.2 Properties of Determinants 3.3 Numerical Evaluation of a Determinant 3.4 Determinants, Matrix Inverses, and Systems of Linear Equations. 3.1 Introduction to Determinants. Definition

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Chapter 3

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  1. Chapter 3 Determinants Gareth Williams J & B, 滄海書局

  2. 3.1 Introduction to Determinants • 3.2 Properties of Determinants • 3.3 Numerical Evaluation of a Determinant • 3.4 Determinants, Matrix Inverses, and Systems of Linear Equations

  3. 3.1 Introduction to Determinants Definition The determinant of a 2  2 matrix A is denoted |A| and its given by Observe that the determinant of a 2  2 matrix is given by the different of the products of the two diagonals of the matrix. The notation det(A) is also used for the determinant of A.

  4. Example 1

  5. Definition Let A be a square matrix. The minor of the element aij is denoted Mij and is the determinant of the matrix that remains after deleting row i and column j of A. The cofactor of aij is denoted Cij and is given by Cij = (–1)i+jMij Note that the minor and cofactor differ in at most sign. Cij = Mij.

  6. Example 2 Determine the minors and cofactors of the elements a11 and a32 of the following matrix A. Solution

  7. Definition The determinant of a square matrix is the sum of the products of the elements of the first row and their cofactors. These equations are called cofactor expansions of |A|.

  8. Example 3 Evaluate the determinant of the following matrix A. Solution

  9. Theorem 3.1 The determinant of a square matrix is the sum of the products of the elements of any row or column and their cofactors. ith row expansion: jth column expansion:

  10. Find the determinant of the following matrix using the second row. Example 4 Solution

  11. Example 5 Evaluate the determinant of the following 4  4 matrix. Solution

  12. Example 6 Solve the following equation for the variable x. Solution

  13. Computing Determinants of 2  2 and 3  3 Matrices

  14. Alternative Definition of Determinant Definition Let A be an n n matrix. Then  Means that the terms are to be summed aver all permutations j1j2…jn of the set {1, 2, …, n}. The sign + or – is selected for each term according to whether the permutation is even or odd.

  15. Alternative Definition of Determinant Example 4 Let A be 2  2 matrix. Cofactor definite: Permutation definition: The permutations of {1, 2} are are 12 (even) and 21(odd). Thus this definition gives The two definition agree for 2  2 matrices.

  16. 3.2 Properties of Determinants Theorem 3.2 • Let A be an n n matrix and c be a nonzero scalar. • If a matrix B is obtained from A by multiplying the elements of a row (column) by c then |B| = c|A|. • If a matrix B is obtained from A by interchanging two rows (columns) then |B| = –|A|. • If a matrix B is obtained from A by adding a multiple of one row (column) to another row (column), then |B| = |A|. Proof

  17. Theorem 3.2

  18. Example 2 If then it can be shown, by expanding in terms of cofactors, that |A| = 12. Use this information, together with the above row and column properties of determinants, to evaluate the determinants of the following matrices. Solution

  19. Singularity and Nonsingularity Definition A square matrix A is said to be singular if |A|=0. A is nonsingular if |A|≠0.

  20. Theorem 3.3 • Let A be a square matrix. A is singular if • All the elements of a row (column) are zero. • two rows (columns) are equal. • two rows (columns) are proportional. Proof

  21. Theorem 3.3

  22. Example 3 Show that the following matrices are singular. Solution

  23. Theorem 3.4 • Let A and B be n n matrices and c be a nonzero scalar. • Determinant of a scalar multiple: |cA| = cn|A|. • Determinant of a product: |AB| = |A||B|. • Determinant of a transpose: |At| = |A|. • Determinant of an inverse: (assuming A–1 exists) Proof

  24. Theorem 3.4

  25. Example 4 If A is a 2  2matrix with |A| = 4, use Theorem 3.4 to compute the following determinants. (a) |3A| (b) |A2| (c) |5AtA–1|, assuming A–1 exists Solution

  26. Example 5 Prove that |A–1AtA| = |A| Proof

  27. Example 6 Prove that if A and B are square matrices of the same size, with A being singular, then AB is also singular. Is the converse true? Proof

  28. 3.3 Numerical Evaluation of a Determinant Definition A square matrix is called an upper triangular matrix if all the elements below the main diagonal are zero. It is called a lower triangular matrix if all the elements above the main diagonal are zero.

  29. The determinant of a triangular matrix is the product of its diagonal elements. Theorem 3.5 Proof

  30. Example 1

  31. Numerical Evaluation of a Determinant • Adding a multiple of one row to another. This transformation leaves the determinant unchanged. 2. Interchanging two rows of the determinant. This transformation multiples the determinant by -1.

  32. Example 2 Evaluation the determinant. Solution

  33. Example 3 Evaluation the determinant. Solution

  34. Example 4 Evaluation the determinant. Solution

  35. Example 5 Evaluation the determinant. Solution

  36. 3.4 Determinants, Matrix Inverse, and Systems of Linear Equations Definition Let A be an n n matrix and Cijbe the cofactor of aij. The matrix whose (i, j)th element is Cij is called the matrix of cofactor of A. The transpose of this matrix is called the adjoint of A and is denoted adj(A).

  37. Example 1 Give the matrix of cofactors and the adjoint matrix of the following matrix A. Solution

  38. Theorem 3.6 Let A be a square matrix with |A|  0. A is invertible with Proof

  39. Theorem 3.7 A square matrix A is invertible if and only if |A|  0. Proof

  40. Example 2 Use a determinant to find out which of the following matrices are invertible. Solution

  41. Example 3 Use the formula for the inverse of a matrix to compute the inverse of the matrix Solution

  42. Theorem 3.8 Let AX = B be a system of n linear equations in n variables. If |A|  0, there is a unique solution. If |A| = 0, there may be many or no solutions. Proof If |A|  0, we know that A–1 exists and there is then a unique solution given by X = A–1B. If |A| = 0, the determinant of every matrix leading up to, and including, the reduced echelon from A is zero. Thus the reduced echelon form of A is not In. The solution to the system AX = B is not unique. The following two systems of linear equations, each of which has a singular matrix of coefficients, illustrate that there may be many or no solutions.

  43. Example 4 Determine whether or not the following system of equations has an unique solution. Solution

  44. Theorem 3.9 Cramer’s Rule1 Let AX = B be a system of n linear equations in n variables such that |A|  0. The system has a unique solution given by Where Ai is the matrix obtained by replacing column i of A with B. Proof

  45. Example 5 Solving the following system of equations using Cramer’s rule. Solution

  46. Example 6 Determine values of  for which the following system of equations has nontrivial solutions.Find the solutions for each value of . Solution

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