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Lecture 5 Determinants

Lecture 5 Determinants. Last Time Elementary Matrices Determinant of a Matrix Evaluation of Determinant Reading Assignment: Secs 2.4, 3.1, 3.2 of Textbook. Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE 翁慶昌 -NTUEE SCC_10_2007. Lecture 5: Determinants. Today

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Lecture 5 Determinants

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  1. Lecture 5 Determinants Last Time Elementary Matrices Determinant of a Matrix Evaluation of Determinant Reading Assignment: Secs 2.4, 3.1, 3.2 of Textbook Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_10_2007

  2. Lecture 5: Determinants Today • Matrix Computation Packages (Matlab and …) • Evaluation of Determinants • Geometric Interpretations • Properties of Determinants Reading Assignment: Secs 3.2, 3.3, 3.5 of Textbook Homework #2 due Next Time • Introduction to Eigenvalues • Applications of Determinants • Vectors in Rn • Vector Spaces Reading Assignment: Secs 3.4, 4.1-4.2

  3. Lecture 4: Elementary Matrices & Determinants Today • Matrix Computation Packages (Matlab and …) • Evaluation of Determinants • Geometric Interpretations • Properties of Determinants

  4. Keywords in Section 2.4: • row elementary matrix: 列基本矩陣 • row equivalent: 列等價 • lower triangular matrix: 下三角矩陣 • upper triangular matrix: 上三角矩陣 • LU-factorization: LU分解

  5. What Did You Actually Learn about Determinant? • Q1: Why can every invertible matrix be represented by mutiplications of elementary matrices?

  6. What Did You Actually Learn about Determinant? Q2: How is determinant related to systems of linear equations?

  7. Thm 3.1: (Expansion by cofactors) Let A isasquare matrix of order n. Then the determinant of A is given by (Cofactor expansion along the i-th row, i=1, 2,…, n) or (Cofactor expansion along the j-th row, j=1, 2,…, n )

  8. Ex: The determinant of a matrix of order 3

  9. The determinant of a matrix of order 3: Subtract these three products. Add these three products.

  10. Upper triangular matrix: All the entries below the main diagonal are zeros. • Lower triangular matrix: All the entries below the main diagonal are zeros. • Diagonal matrix: All the entries above and below the main diagonal are zeros.

  11. Keywords in Section 3.1: • determinant : 行列式 • minor : 子行列式 • cofactor : 餘因子 • expansion by cofactors : 餘因子展開 • upper triangular matrix: 上三角矩陣 • lower triangular matrix: 下三角矩陣 • diagonal matrix: 對角矩陣

  12. Geometry of Determinants: Determinants as Size Functions • We have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of that determinant. One way to compute the area that it encloses is to draw this rectangle and subtract the area of each subregion.

  13. The region formed by and is bigger, by a factor of k, than the shaded region enclosed by and . That is, size ( , ) = k ·size( , ) and in general we expect of the size measure that size(. . . , , . . . ) = k ·size(. . . , , . . . ). • The properties in the definition of determinants make reasonable postulates for a function that measures the size of the region enclosed by the vectors in the matrix. See this case:

  14. Another property of determinants is that they are unaffected by pivoting. Here are before-pivoting and after-pivoting boxes (the scalar used is • k = 0.35). Although the region on the right, the box formed by and , is more slanted than the shaded region, the two have the same base and the same height and hence the same area. This illustrates that

  15. That is, we’ve got an intuitive justification to interpret det ( , . . . , ) as the size of the box formed by the vectors. Example The volume of this parallelepiped, which can be found by the usual formula from high school geometry, is 12.

  16. The only difference between them is in the order in which the vectors are taken. If we take first and then go to , follow the counterclockwise are shown, then the sign is positive. Following a clockwise are gives a negative sign. The sign returned by the size function reflects the ‘orientation’ or ‘sense’ of the box.

  17. Volume, because it is an absolute value, does not depend on the order in which the vectors are given. The volume of the parallelepiped in the following example, can also be computed as the absolute value of this determinant. The definition of volume gives a geometric interpretation to something in the space, boxes made from vectors.

  18. 3.2 Evaluation of a determinant using elementary operations • Thm 3.3: (Elementary row operations and determinants) Let A and B be square matrices.

  19. Ex:

  20. Sol: Note: A row-echelon form of a square matrix is always upper triangular. • Ex 2: (Evaluation a determinant using elementary row operations)

  21. Notes:

  22. Notes:

  23. Thm 3.4: (Conditions that yield a zero determinant) If A is a square matrix and any of the following conditions is true, then det (A) = 0. (a) An entire row (or an entire column) consists of zeros. (b) Two rows (or two columns) are equal. (c) One row (or column) is a multiple of another row (or column).

  24. Ex:

  25. Note: Number of operations for cofactor expansion of nxn matrix ~ n! 30! = ?

  26. Ex 5: (Evaluating a determinant) Sol:

  27. Ex 6: (Evaluating a determinant) Sol:

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