Determinants of Matrices | Lecture Note

University:
Montgomery College
Course:
MATH 120: Survey of College Mathematics
Academic year:

2024
Views:

44

Pages:

2
Author:

Alvin Bradford

Determinants of Matrices Determinants are a property of square matrices. The determinant of A is written |A|. The determinant of a 2  2 matrix A  a b c d is |A|  a  d – b  c For higher rank matrices, we can use cofactors to calculate their determinants: The cofactor of an element in row I and column J is the determinant of the matrix that remains after row I and column J are removed: column 3 A11 A12 A13 A14 A15 A21 A22 A23 A24 A25 row 4 Here the cofactor for A43 is A11 A12 A14 A15 A21 A22 A24 A25 A31 A32 A33 A34 A35 A31 A32 A34 A35 A41 A42 A43 A44 A45 A51 A52 A54 A55 A51 A52 A53 A54 A55    A11 A12 A13 To calculate the determinant of a 3  3 matrix: A21 A22 A23 Write alternating ’s and ’s above the top row A31 A32 A33 Multiply each element in the top row by its cofactor. If a  is above the column, then add it. If a is above the column, then subtract it. So |A|  A11 A22 A23 A32 A33  A12 A21 A23 A31 A33  A13 A21 A22 A31 A32 Or |A|  A11 (A22 A33  A23 A32)  A12 (A21 A33  A23 A31)  A13 (A21 A32  A22 A31) We can also write |A|  A11 A22 A33  A12 A23 A31  A13 A21 A32  A13 A22 A31  A12 A21 A33  A11 A23 A32 Which we can also get by multiplying and adding all terms on lines starting at the upper left and going to the lower right, and then multiplying and subtracting all terms on lines starting at the lower left and going to the upper right. A diagram may help: A11 A12 A13 A21 A22 A23 A31 A32 A33 A12 A23 A31 A13 A21 A32  A11 A23 A32 A11 A12 A13 A21 A22 A23 A31 A32 A33  A13 A22 A31  A12 A21 A33 A11 A22 A33 Note: We can use the elements in any row to compute the determinant using cofactors. Use a grid of alternating ’s and ’s to determine whether to a term should be added or subtracted    For example, given the 3  3 matrix A above, the grid would be So |A| = A21 A12 A13  A22 A32 A33 = A31 A12 A13 A22 A23 A11 A13  A23 A31 A33  A32 A11 A13 A21 A23 A11 A12 A31 A32  A33 A11 A12 A21 A22      