Week3

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Week3

  1. 1. a b Definition: For a 2 2 matrix A c d define its determinant by: det(A) = |A| = ad bc Observe that det(A) is a scalar that in a way summarizes the whole matrix A.
  2. 2. Definition: a11 a12 a13 The determinant A a21 a22 a23 of a 3 3 matrix: a31 a32 a33 is defined by: |A| = a11 a12 a13 a22 a23 a21 a23 a21 a22 a21 a22 a23 a11 a12 a13 a32 a33 a31 a33 a31 a32 a31 a32 a33
  3. 3. Let A be an n n matrix, define Mij to be the (i,j)-minor of A, i.e. the resulting matrix after removing row i and column j from A Also define Cij = ( 1)i+jdet(Mij) to be the (i,j)-cofactor of A.
  4. 4. Then, the determinant of A can be computed by: det(A) = j aijCij = ai1Ci1 + ai2Ci2 + + ainCin (a cofactor expansion along the ith row) or by: det(A) = i aijCij = a1jC1j + a2jC2j + + anjCnj (a cofactor expansion along the jth column).
  5. 5. Find the determinant of the matrix 3 1 -4 A= 2 5 6 1 4 8 using the first row using the second column
  6. 6. 5 6 2 6 2 5 (A ) 3 1 4 4 8 1 8 1 4 = 3 (1 6 )-1 0 -4(3 )= 2 6
  7. 7. 2 6 3 4 3 4 (A ) 1 5 4 1 8 1 8 2 6 =-10+5(28)-4(26) =26
  8. 8. 1. If A has a zero row or column, then |A| = 0. 2. If A is upper or lower triangular matrix, then |A| = a11a22 ann. 3. If A is a diagonal matrix, then |A| = a11a22 ann. 4. |In| = 1
  9. 9. 5- If B is obtained by switching two rows (or columns) of A, then |B| = |A|. 6- If B is obtained by multiplying a row (or a column) of A by k, then |B| = k|A|. 7- If B is obtained by adding a multiple of a row (or a column) of A to another row (column), then |B| = |A|.
  10. 10. 8- |A| = |AT| 9- If two rows (columns) are identical then |A| = 0 10- |AB| = |A| |B| if A and B are of the same order. 11- |kA| = kn |A|
  11. 11. A mxn matrix can be written as R1 A= R2 . . Rm Ri=[ai1,ai2, ,ain] row i of A Also we can write A as A=[C1,C2, ,Cn] where Cj is column j of A
  12. 12. 2 4 8 A 3 6 12 1 5 9 1 2 4 1 2 4 =2 3 6 12 2x 3 1 2 4 0 1 5 9 1 5 9
  13. 13. 6 4 5 B 5 4 6 1 0 1 1 0 1 R R1 2 5 4 6 0 1 0 1
  14. 14. 12 9 3 4 3 2 A 0 5 4 0 5 4 4 3 2 12 9 3 4 3 2 3R 1 R 3 0 5 4 (4)(5)( 3) 60 0 0 3
  15. 15. A row Rs is said to be a linear combination of R1,R2, ,Rm if there exist real numbers k1,k2, ,km such that Rs = k1R1+k2R2+ +kmRm
  16. 16. For the matrix A, defined below, show that R2 can be written as a linear combination of the rows of A 1 3 2 4 3 5 0 7 A 2 1 5 2 3 0 1 1 R2=R4-R3+2R1
  17. 17. For the matrix A, defined below, show that C3 can be written as a linear combination of the columns of A 1 2 3 A 2 3 5 2 2 4 C 3 C1 C 2
  18. 18. If a row (column) of a matrix A can be expressed as a linear combination of the other rows (columns) we say that the rows (columns) of A are linearly dependent
  19. 19. The rows of a matrix A are linearly independent if the only solution of k1R1+k2R2+ +kmRm=0 is k1=k2= =km=0 i.e. any row cannot be written as a linear combination of the other rows
  20. 20. If the rows (columns) of A are linearly dependent then
  21. 21. Use the determinates properties to show that (A) = 0 2 1 1 A 4 1 5 12 3 9 C 1 C 2 C 3
  22. 22. A square matrix A is invertible if and only if (A) 0
  23. 23. A square matrix A is invertible if and only if its rows (columns) are linearly independent
  24. 24. If A is invertible then |A-1| = 1/|A| Proof: Since A is invertible, then AA-1=In |AA-1| = |In| = 1 |AA-1| = |A| |A-1| =1 Since |A| 0, then |A-1| = 1/|A|
  25. 25. Let A be an n n square matrix. The following statements are all equivalent: 1. 2. 3. 4.
  26. 26. Find all values of k, for which the following matrix is invertible: k 2 2 A 2 k 2 2 2 k
  27. 27. If k=2 then A =0 k C1 C 2 k k k
  28. 28. k k A k k k 2 k k 2 3
  29. 29. Show that x=3 is one of the roots of the equation
  30. 30. Show that the matrix 1 2 3 A 1 0 1 2 4 6 is not invertible 2R2=R1 A =0
  31. 31. A non-zero matrix A is said to have rank k r(A) = k if at least one of its k-square minors is different from zero while every (k+1)- square minors, if any, is zero. A zero matrix is said to have rank zero.
  32. 32. mxn
  33. 33. An n-square matrix is said to be full rank matrix if r(A) = n. Result: The n-square matrix A is invertible if and only if r(A) = n
  34. 34. Find the rank of A = 2 1 1 4 1 5 12 3 9 C2=C1-C3 A =0 M33 = 2 1 6 0 4 1 r(A) =2
  35. 35. Find the rank of A = 1 2 3 5 10 15 2 4 6 R3 = 2 R1 R2 = -5 R1 r(A) = 1 Note that A =0 and all 2x2 minors are zero also.
  36. 36. The following operations, called elementary transformations on a matrix do not change either its order or its rank: 1- Interchanging two rows (columns) 2- The multiplication of every element of of row (column) by a nonzero constant k.
  37. 37. 3- The multiplication of every element of a row (column) by a nonzero constant k and adding the result to another row (column).
  38. 38. Two matrices A and B are called equivalent, A B , if one can be obtained from the other by a sequence of elementary transformations.
  39. 39. Equivalent matrices have the same order and the same rank.
  40. 40. 1 2 1 A 2 4 3 2R1 R 2 1 2 1 1 2 1 0 0 5 R1 R 3 1 2 1 1 2 1 0 0 5 0 0 0
  41. 41. Show that the following matrix A is equivalent to the identity matrix I2 2 2 A 1 4
  42. 42. 1 1 1 R1 A 2 1 4 1 1 R1 R 2 A 0 3 1 1 1 R 2 A I 2 3 0 1
  43. 43. Given an n-square matrix A, the following statements are equivalent: 1- A is invertible. 2- r(A) = n. 3- A In 4- A 0 5- All rows of A are linearly independent.

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