Matrix Determinants and Cofactor Expansion
Discover the concept of determinants in square matrices, their applications, and how to compute them using cofactor expansion method. Explore examples and learn about tridiagonal matrices. Dive into the world of matrix algebra!
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Presentation Transcript
Determinant Hung-yi Lee The determinant of a square matrix is a scalar that provides information about the matrix (e.g. invertible or not)
Determinants in High School 2 X 2 3 x 3 ?1 ?4 ?7 ?2 ?5 ?8 ?3 ?6 ?9 ? =? ? ? ? = ? ??? ? = ??? ? = ?? ?? ?1?5?9+?2?6?7+?3?4?8 ?3?5?7 ?2?4?9 ?1?6?8
Cofactor Expansion aij: scalar Aij: matrix Suppose A is an n x n matrix. Aij is defined as the submatrix of A obtained by removing the i-th row and the j-th column. A Aij i-th row (n-1) x (n-1) J-th column
?1? ?2? ??? ?11 ?21 ??1 ?12 ?22 ??2 Cofactor Expansion ? = ?11 Pick row 1 ???? = ?11?11+ ?12?12+ + ?1??1? Or pick row i ???? = ??1??1+ ??2??2+ + ?????? cij: (i,j)-cofactor Or pick column j ???? = ?1??1?+ ?2??2?+ + ?????? ???= 1?+??????? Cofactor expansion again ?11= 11+1????11
???? = ?11?11+ + ?1??1? ???= 1?+??????? For 1x1 matrix, det([a]) = a Turtles all the way down?
2 x 2 matrix ???= 1?+??????? Define det([a]) = a ? =? ? ? ??? ? = ?? ?? ? Pick the first row ??? ? = ??11+ ??12 = ?? ?? ?11= 11+1??? ? = ? ?12= 11+2??? ? = ?
3 x 3 matrix ???= 1?+??????? 1 4 7 2 5 8 3 6 9 ? = Pick row 2 ???? = ?21?21+ ?22?22+ ?23?23 4 6 5 12+3????23 12+1????21 12+2????22 1 4 7 2 5 8 3 6 9 1 4 7 2 5 8 3 6 9 1 4 7 2 5 8 3 6 9 ?21= ?22= ?23=
Example Given tridiagonal n x n matrix A 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 ? = Find det A when n = 999
????4 = ?11?11+ ?12?12+ ?13?13+ ?14?14 1 1 ?2=1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 ?11= 12??? ?3= = ??? ?3 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 ?4= ?12= 13??? = ??? ?2 = ?11?11+ ?12?12+ ?13?13 1 1 0 = ??? ?2 ??? ?4 = ??? ?3 ??? ?2
??? ?? = ??? ??1 ??? ??2 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 ??= ?? 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 ? = ?? 2 = 0
Example ??? ?? = ??? ?? 1 ??? ?? 2 ??? ?1 = 1 ??? ?3 = 1 ??? ?2 = 0 ??? ?5 = 0 ??? ?6 = 1 ??? ?4 = 1 ??? ?7 = 1 ??? ?8 = 0
