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ENGG2013 Unit 8 2x2 Determinant

ENGG2013 Unit 8 2x2 Determinant. Jan, 2011. Last time. Invertible matrix (a.k.a. non-singular matrix) Represents reversible linear transformation Gauss-Jordan elimination Algorithm for compute matrix inverse. Carl Friedrich Gauss (1777 – 1853). Borned: Braunschweig, Germany.

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ENGG2013 Unit 8 2x2 Determinant

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  1. ENGG2013 Unit 82x2 Determinant Jan, 2011.

  2. Last time • Invertible matrix (a.k.a. non-singular matrix) • Represents reversible linear transformation • Gauss-Jordan elimination • Algorithm for compute matrix inverse ENGG2013

  3. Carl Friedrich Gauss (1777 – 1853) • Borned: Braunschweig, Germany. • Died: Göttingen, Germany. • Great mathematician in the 18th century. • Legacy • Gaussian distribution • Gaussian elimination • Gauss-Jordan elimination • Gaussian curvature in differential geometry • Proof of the quadratic reciprocity in number theory • Construction of 17-gon by straightedge and compass • And much more ENGG2013

  4. Wilhelm Jordan (1842 –1899) http://en.wikipedia.org/wiki/Wilhelm_Jordan • Borned: Ellwangen, Germany. • Died: Hanover, Germany. • Geodesist. • Remembered for: • Surveying in Germany and Africa • His book “Textbook of Geodesy” (HandbuchderVermessungskunde) popularizes the Gauss-Jordan algorithm ENGG2013

  5. 22 Determinant • Area of parallelogram (c,d) Row 1 is the first vector Row 2 is the second vector d (a,b) b a c ENGG2013

  6. 22 Determinant • Area of parallelogram (2,4) Area = 10 Row 1 is the first vector Row 2 is the second vector (3,1) ENGG2013

  7. Outline • Computing 22 matrix inverse via determinant • Properties of determinant • Explain why the absolute value of determinant is the area of parallelogram? ENGG2013

  8. Determinant of 22 matrix • Notation: Given a 22 matrix we use the notation to stand for the determinant ps – qr. or ENGG2013

  9. Example Determinant of identity matrix is 1 1 1 ENGG2013

  10. Example Determinant of a diagonal matrixis the area of a rectangle h w ENGG2013

  11. A formula for the Inverse of a 2x2 matrix • Given 22 matrix • Want to find the inverse of A • A formula for A-1: If det A is nonzero, we have ENGG2013

  12. How to compute the inverseof a 2x2 matrix • Exchange the two diagonal entries a, d. • Take the negative of the two off-diagonal entries b, c. • Divide by the determinant. ENGG2013

  13. Application in solving equations • Solve • If we know the inverse of the 2x2 matrix, we can solve the linear system easily. ENGG2013

  14. Properties of determinant • A matrix and its transpose has the same determinant “The transpose of a matrix” means reflecting the matrixalong the diagonal. Row 1 and row 2 becomecolumn 1 and column 2, andvice versa. We write AT for the transpose of matrix A. Proof is obvious, because ENGG2013

  15. Transposing does not change area (1,4) Area = 10 (2,4) Area = 10 (3,2) (3,1) ENGG2013

  16. Meta-property By property 1, we have: Any row property of determinant is a column property, and vice versa ENGG2013

  17. Properties of determinant • If any row or column is zero, then the determinant is 0. Zero area ENGG2013

  18. Properties of determinant • If the two columns (or two rows) are constant multiple of each other, the determinant is zero. Zero area ENGG2013

  19. Properties of Determinant • If we exchange of the two columns (or two rows), the determinant is multiplied by –1. (1,3) The first kindof elementaryrow operation (2,1) ENGG2013

  20. Properties of Determinant • If we multiply a row (or a column) by a constant c, the value of determinant also increase by a factor of c. (4,4) The 2nd kindof elementaryrow operation (0,1) (1,1) ENGG2013

  21. Properties of Determinant • If we add a constant multiple of a row (column) to the other row (column), the determinant does not change. (0,1) (3,1) The 3rd kindof elementaryrow operation (1,0) ENGG2013

  22. Properties of Determinant • In the linear transformation represented by a 22 matrix, the magnitude of determinant measures the area expanding factor. Multiplied by (1,1) (0,1) (1,0) Square withunit area = ad – bc The ratio of area ENGG2013

  23. Properties of Determinant • For any 2x2 matrices A and B, we have the following multiplicative property Multiplied by A Multiplied by B Parallelogramarea = det(AB)= det(A) det(B) Square withunit area Parallelogramarea = det(A) Expand by a factor of det(B) Expand by a factor of det(A) Expand by a factor of det(AB) = det(A) det(B) ENGG2013

  24. Proof of property 8 • Let ENGG2013

  25. Determinant as area • Using the properties in previous pages, we are now ready to show that the absolute value of det(M) is equal to the area of parallelogram whose sides are the two rows of M. • We divide the argument into two steps • The two rows are perpendicular (special case). • The two rows are not perpendicular (general case). ENGG2013

  26. Determinant as area (I) • Suppose that the two rows are perpendicular i.e., ac+bd = 0 (dot product of [a b] and [c d] are zero) • Let • Want to show that (c,d) (a,b) The trick is to show instead. ENGG2013

  27. Determinant as area (I) By Property 1 Just write down M and MT By Property 8 By the definition ofmatrix multiplication Because the dot productac+bd is zero by assumption By the definition of determinant By Pythagoras theorem, are the height and width of the rectangle. ENGG2013

  28. Determinant as area (II) • Suppose that the two rows of M are not perpendicular. • Idea: “Slide” the parallelogramto a rectangle, whilekeeping the areaunchanged. (c,d) (a,b) ENGG2013

  29. Determinant as Area (II) (c,d) Decompose [c d] into two components, one is perpendicular to [a b],and the other along the same direction as [a b]. (a,b) Perpendicular to [a b](The height of parallelogram) In the same direction as [a b] Find the constant k (hoemwork exercise) ENGG2013

  30. Determinant as area (II) • Choose a constant k such that • Let • [c’ d’] is perpendicular to [ab] by our choice of k. • By definition the area of parallelogram is equal to • But By property 6 By the first part of our proof in p.27 ENGG2013

  31. We have proved the following theorem (see the picture in p.5) Theorem: For 2x2 matrix M, the absolute value of det(M) is equal to the area of the parallelogram whose sides are the two rows (or the two columns) of M. ENGG2013

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