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3.9 Determinants

3.9 Determinants. Given a square matrix A its determinant is a real number associated with the matrix. The determinant of A is written: det( A ) or | A | For a 2x2 matrix, the definition is. a. b. a. b. det = = ad - bc. c. d. c. d.

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3.9 Determinants

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  1. 3.9 Determinants • Given a square matrix Aits determinant is a real number associated with the matrix. • The determinant of A is written: det(A) or |A| • For a 2x2 matrix, the definition is a b a b det = = ad - bc c d c d • For larger matrices the definition is more complicated

  2. -5 2 det = = (-5)(0) – (2)(-2) = 4 -2 0 Determinants 2x2 examples -5 1 1 1 1 2 2 2 2 2 det = = (1)(4) – (2)(3) = -2 -2 3 2 2 3 4 0 4 4 4 det = = (1)(4) – (2)(2) = 0

  3. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M11 : remove row 1, col 1 1 1 -2 A = -1 2 3 2 3 M11 = 2 7 0 7 0

  4. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M12 : remove row 1, col 2 1 1 -2 A = -1 2 3 -1 3 M12 = 2 7 0 2 0

  5. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M13 : remove row 1, col 3 1 1 -2 A = -1 2 3 -1 2 M13 = 2 7 0 2 7

  6. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M21 : remove row 2, col 1 -2 1 1 A = -1 2 3 1 -2 M21 = 2 7 0 7 0

  7. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M22 : remove row 2, col 2 -2 1 1 A = -1 2 3 1 -2 M22 = 2 7 0 2 0

  8. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M23 : remove row 2, col 3 -2 1 1 A = -1 2 3 1 1 M23 = 2 7 0 2 7

  9. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M31 : remove row 3, col 1 -2 1 1 A = -1 2 3 1 -2 M31 = 2 7 0 2 3

  10. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M32 : remove row 3, col 2 -2 1 1 A = -1 2 3 1 -2 M32 = 2 7 0 -1 3

  11. Determinants • To define det(A) for larger matrices, we will need the definition of a minor Mij • The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A M33 : remove row 3, col 3 -2 1 1 A = -1 2 3 1 1 M33 = 2 7 0 -1 2

  12. 3.9.1 The formula for a 3x3 matrix a13 a11 a12 • For a matrix a21 a23 A = a22 a32 a31 a33 • Its determinant is given by |A| = a11|M11| - a12|M12| + a13|M13| • From the formula for a 2x2 matrix: a22 a23 |M11|= = a22a33 - a23a32 a32 a33

  13. 3.9.1 The formula for a 3x3 matrix a13 a11 a12 • For a matrix a21 a23 A = a22 a32 a31 a33 • Its determinant is given by |A| = a11|M11| - a12|M12| + a13|M13| • From the formula for a 2x2 matrix: a21 a23 |M12|= = a21a33 - a23a31 a31 a33

  14. 3.9.1 The formula for a 3x3 matrix a13 a11 a12 • For a matrix a21 a23 A = a22 a32 a31 a33 • Its determinant is given by |A| = a11|M11| - a12|M12| + a13|M13| • From the formula for a 2x2 matrix: a21 a22 |M13|= = a21a32 - a31a22 a31 a32

  15. 3x3 Example |A| = 1x|M11| - 1x|M12| + (-2)x|M13| 1 1 -2 -1 3 2 -1 3 2 |A|= 1x - 1x + (-2) A = -1 2 3 2 0 7 2 0 7 2 7 0 = 1x(-21) -1x(-6) +(-2)x(-11) = 7

  16. 3x3 Example |B| = 0x|M11| - 1x|M12| + 3x|M13| 0 1 3 5 1 3 5 1 3 |B|= 0x - 1x + 3 x B = 5 3 1 -1 0 2 -1 0 2 -1 2 0 = 0x(-2) -1x(1) +(3)x(13) = 38

  17. 3.9.1 The formula for a 3x3 matrix a13 a11 a12 • For the matrix a21 a23 A = a22 a32 a31 a33 • We used the top row to calculate the determinant: |A| = a11|M11| - a12|M12| + a13|M13| • However, we could equally have used any row of the matrix and performed a similar calculation

  18. 3.9.1 The formula for a 3x3 matrix a13 a11 a12 • For the matrix a21 a23 A = a22 a32 a31 a33 • Using the top row: |A| = a11|M11| - a12|M12| + a13|M13| • Using the second row |A| = -a21|M21| + a22|M22| - a23|M23| • Using the third row |A| = a31|M31| - a32|M32| + a33|M33|

  19. 3.9.1 The formula for a 3x3 matrix |A| = a11|M11| - a12|M12| + a13|M13| + = -a21|M21| + a22|M22| - a23|M23| = a31|M31| - a32|M32| + a33|M33| • Notice the changing signs depending on what row we use: - + - - + + - +

  20. 3.9.1 The formula for a 3x3 matrix a13 + a11 a12 • Equally, we could have used any column as long as we follow the signs pattern a21 a23 A = a22 a32 a31 a33 • E.g. using the first column: - + |A| = a11|M11| - a21|M21| + a31|M31| - - + + - +

  21. This choice sometimes makes it a bit easier to calculate determinants. e.g. • Using the first row: 1 1 -2 0 3 2 0 3 2 |A|= 1x - 1x + (-2) x A = 0 2 3 0 1 1 0 1 1 0 1 1 = 1x(-1) -1x(0) + (-2)x(0) = -1

  22. This choice sometimes makes it a bit easier to calculate determinants. e.g. • However, using the first column: 1 1 -2 2 3 |A|= 1x - 0 + 0 = 1x(-1) = -1 A = 0 2 3 1 1 0 1 1

  23. 3.9.2 A general formula for determinants • For a 4x4 matrix we add up minors like the 3x3 case, and again use the same signs pattern - + - + - - + + + - + - - - + + • Notice that if we think of the signs pattern as a matrix, then it can be written as (-1)i+j

  24. 3.9.2 A general formula for determinants • For a nxn matrix A=(aij)the co-factors of A are defined by Cij:= (-1)i+j|Mij| • The determinant of A is given by the formula |A|= aijCijfor any j=1,2,...,n • Or, |A|= aijCijfor any i=1,2,...,n

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