Determinants. Artem Los February 6th, Artem Los Determinants February 6th, / 16

Size: px

Start display at page:

Download "Determinants. Artem Los February 6th, Artem Los Determinants February 6th, / 16"

Madeleine Collins
5 years ago
Views:

1 Determinants Artem Los February 6th, 2017 Artem Los Determinants February 6th, / 16

2 Overview 1 What is a Determinant? 2 Rules and Theorems 3 Area of a Parallelogram Artem Los (arteml@kth.se) Determinants February 6th, / 16

3 What is a Determinant? Artem Los (arteml@kth.se) Determinants February 6th, / 16

4 Definition Determinant In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. (from For example, Deducing if matrix A is invertible, i.e. A 1 Finding area of a parallelogram Calculating eigenvalues (next week) Artem Los (arteml@kth.se) Determinants February 6th, / 16

5 Computing determinant The neat property of determinants is that we can always express a determinant of matrix n n using determinants of size (n 1) (n 1). Base case. 2 2 a b c d = ad bc Increasing dimension. 3 3 a b c d e f j h i = a e h a b c d e f j h i = a e h f i b d j f i d b h f i + c d j c i + j b e e h c f Artem Los (arteml@kth.se) Determinants February 6th, / 16

6 Determine sign of the coefficients Question. In the example below, why is a positive but b negative? a b c d e f j h i = a e f h i b d f j i + c d e j h For 3 3, we have that: In general, the coefficient is ( 1) i+j for element in ith row and jth column Artem Los (arteml@kth.se) Determinants February 6th, / 16

7 Example Problem. Compute the determinant of A, defined as: A = Step 1: Golden rule of determinant calculations: be lazy. Pick the row/column with most zeros. Why? A = = = 6(3 2) = 6 Artem Los (arteml@kth.se) Determinants February 6th, / 16

8 Rules and Theorems Artem Los Determinants February 6th, / 16

9 Rules (idea) Idea. Determinant matrix reductions are carried out to make it easier to figure out the determinant. The rules differ from elementary row operations. Goal. Get as many zeros as possible, for then we get less terms to compute. Artem Los Determinants February 6th, / 16

10 Rules (continued) Constant term factorization. det B = c det A c 1 c 2 c = c Swapping two rows. det B = det A = Adding two rows/columns. det B = det A = {R : R 2 R 1 } = Artem Los (arteml@kth.se) Determinants February 6th, / 16

11 Theorems Theorem. Let A be a square matrix, i.e. n n. Then: det A = det A T det A 1 = 1 det A det AB = det A det B If two rows are equal = det A = 0 Theorem. Given that A is a square matrix, i.e. n n, the following statements are equivalent: det A 0 rank(a) = n A is invertible Artem Los (arteml@kth.se) Determinants February 6th, / 16

12 Example Problem. Find det A T A given that A is defined as shown below: A = Step 1: (using known theorem): det (A T A) = (det A) 2. Step 2: Find det A = = = = = = ( ) = 13 Artem Los (arteml@kth.se) Determinants February 6th, / 16

13 Area of a Parallelogram Artem Los (arteml@kth.se) Determinants February 6th, / 16

14 Finding area given two spanning vectors y u Area v x Given a vectors u, v that span a parallelogram, the area is given by: ( Area = det u1 u 2 v 1 v 2) In general, if vectors u, v don t lie in the same plane (eg. u, v R 3 ) i j k Area = u v = det u 1 u 2 u 3 v 1 v 2 v 3 Artem Los (arteml@kth.se) Determinants February 6th, / 16

15 Example Problem. Given that a parallelogram is spanned by vectors u = (1, 2) and v = (3, 4), find its area. Step 1: Plug vectors into a matrix: = = 2 Step 2: Take the absolute value, since it s an area: Area is equal to 2 area units. 2 = 2 Artem Los (arteml@kth.se) Determinants February 6th, / 16

16 Volume of parallelepiped Given vectors u, v, w R 3, the volume of the parallelepiped that they span up is: u 1 u 2 u 3 Volume = w ( u v) = det v 1 v 2 v 3 w 1 w 2 w 3 Although the geometrical fact is useful, keep in mind the trick of converting w ( u v) to the determinant form. Artem Los (arteml@kth.se) Determinants February 6th, / 16

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij