MAT211_Lecture17[1] - Recall that a 2 x 2 matrix A a b d...

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MAT211 Lecture 17 Determinants. Recall that a 2 x 2 matrix A is invertible If and only if a.d-b.c ! 0. The number a.d-b.c is the the determinant of A. a b c d Definition A pattern in an n x n matrix A is a choice of entries of A, such that there is only one entry in each row and only one in each column. Example Find all possible patterns in 2x2 matrices Find all possible patterns in 3x3 matrices. How many patterns are there in 4x4 matrices? And in n x n matrices? Definition The entries of a pattern P of a matrix A are inverted if one of them is located right and above the other in A. The signature of a pattern P of a matrix A, denoted by sgn(P) is (-1) (number of pairs of inverted entries in P) EXAMPLE Find the signature of the pattern of a 3 x 3 matrix A, a 31 , a 22 , a 13
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Definition The determinant a square matrix A, denoted by det A is " sgn(P).product(elements in P) where the sum is taken over all patterns P.
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