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ENGG2013 Unit 9 3x3 Determinant

ENGG2013 Unit 9 3x3 Determinant. Feb, 2011. Last time. 2 2 determinant Compute the area of a parallelogram by determinant A formula for 2x2 matrix inverse. Today. 3 3 determinant and its properties Using determinant, we can test whether three vectors lie on the same plane

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ENGG2013 Unit 9 3x3 Determinant

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  1. ENGG2013 Unit 93x3 Determinant Feb, 2011.

  2. Last time • 22 determinant • Compute the area of a parallelogram by determinant • A formula for 2x2 matrix inverse ENGG2013

  3. Today • 33 determinant and its properties • Using determinant, we can • test whether three vectors lie on the same plane • solve 33 linear system • test whether the inverse of a 33 matrix exists ENGG2013

  4. Vector Notation • We will use two different notations for a point in the 3D space (x,y,z) z z y y x x ENGG2013

  5. 22 determinant Notation for 22 determinant : + – How to calculate: ad – bc - bc ad ENGG2013

  6. 33 determinant Notation for 33 determinant : Definition: ENGG2013

  7. Rule of Sarrus – – + + + – Pierre Frédéric Sarrus (1798 – 1861) ENGG2013

  8. Volume of parallelepiped • Geometric meaning • The magnitude of 33 determinant is the volume of a parallelepiped z y x ENGG2013

  9. Co-planar  zero determinant • Determinant = 0 Volume = 0  the three vectors lie on the same plane z y A collection of vectorsare said to be co-planarif they lie on the same plane. x ENGG2013

  10. Det of Diagonal matrix • Volume of a rectangular box c b a ENGG2013

  11. Transpose has the same determinant – – + + + – Compare with ENGG2013

  12. Volume of parallelepiped • In computing the volume of a parallelepiped, it does not matter whether we write the vector horizontally or vertically in the determinant z Volume of parallelepiped with vertices(0,0,0), (1,2,0), (2,0,1), (–1, 1, 3) equals tothe absolute value of y or x ENGG2013

  13. Question • Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on the same plane? ENGG2013

  14. Cramer’s rule • If the determinant of a 33 matrix A is non-zero, we can solve the linear system Ax = b by Cramer’s rule. • The solution to is or equivalently A x b Gabriel Cramer (1704-1752) ENGG2013

  15. PROPERTIES OF DETERMINANT ENGG2013

  16. How to show that Cramer’s rule does give the correct answer? • The Cramer’s rule is a theorem, which requires a proof, or verification. • We need some properties of determinant. ENGG2013

  17. Properties of determinant • Taking transpose does not change the value of determinant We have already verified this property in p.11 ENGG2013

  18. Meta-property • Because • After taking the transpose of a matrix, columns become rows, and rows become column. • Taking the transpose of a matrix does not change the value of its determinant. • Therefore, any row property of determinant is automatically a column property, and vice versa. ENGG2013

  19. Properties of determinant • If any row or column is zero, then the determinant is 0. For example ENGG2013

  20. Properties of determinant • If any two columns (or rows) are the identical, then the determinant is zero. For example, if the second column and the third column are the same, then ENGG2013

  21. Properties of Determinant • If we exchange of the two columns (or two rows), the determinant is multiplied by –1. For example, if we exchange the column 2 and column 3, we have The first kindof elementaryrow operation ENGG2013

  22. Multiply by a constant • If we multiply a row (or a column) by a constant k, the value of determinant increases by a factor of k. For example, if we multiply the third row by a constant k, The 2nd kindof elementaryrow operation ENGG2013

  23. An additive property • If a row (or column) of a determinant is the sum of two rows (or columns), the determinant can be split as the sum of two determinants For example, if the first column is the sum of two column vectors, thenwe have ENGG2013

  24. Properties of Determinant • If we add a constant multiple of a row (column) to the other row (column), the determinant does not change. For example, if we replace the 3rd column by the sum of the 3rd column and k times the 2nd column, The 3rd kindof elementaryrow operation ENGG2013

  25. Summary on the effect of theelementary row (or column) operations on determinant • Exchange two rows (or columns) change the sign of determinant • Multiply a row (or a column) by a constant k  multiply the determinant by k • Add a constant multiple of a row (column) to another row (or column)  no change ENGG2013

  26. Proof of the Cramer’s rule • The solution to is Verification for x1: Substitute the value of b1, b2 and b3 in the first column of A. Verification for x2: Substitute the value of b1, b2 and b3 in the second column of A. Etc. Cramer’s rule in wikipedia ENGG2013

  27. Because x1, x2, x3 satisfy the system of linear equations, we have By substitution Property 6 Property 5 =0 =0 By Property 3 ENGG2013

  28. MINOR AND COFACTOR ENGG2013

  29. Another way to compute det Group the six terms as 33 determinant can be computed in terms of 22 determinant ENGG2013

  30. Minor and cofactor • A minor of a matrix is the determinant of some smaller square matrix, obtained by removing one or more of its rows and columns. • Notation: Given a matrix A, the minor obtained by removing the i-th row and j-th column is denoted by Aij. It is also called the minor of the (i,j)-entry aij in A. ENGG2013

  31. Expansion on the first row Minor of a1 Minor of b1 Minor of c1 ENGG2013

  32. Expansion on the second row Minor of a2 Minor of b2 Minor of c2 ENGG2013

  33. Expansion on the third row Minor of a3 Minor of b3 Minor of c3 ENGG2013

  34. The sign pattern Expansion on the first row Expansion on the second row Expansion on the last row ENGG2013

  35. Column expansion We have similar recursive formula fordeterminant by column expansion For example, Computation on the third column is easy, because there are lots of zeros. ENGG2013

  36. Cofactor • The minor together with the appropriate  sign is called cofactor. • For The cofactor of Cijis defined as Expansion on the i-th row (i=1,2,3): Expansion on the j-th column (j=1,2,3): The sign The minor of aij ENGG2013

  37. A formula for matrix inverse Suppose that det A is nonzero. (Beware of thesubscripts) Usually called the adjoint of A Three steps in computing above formula1. for i,j = 1,2,3, replace each aij by cofactor Cij2. Take the transpose of the resulting matrix.3. divide by the determinant of A. ENGG2013

  38. A Quotation Algebra is but written geometry; geometry is but drawn algebra. --- Sophie Germain (1776-1831) L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée ENGG2013

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