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Notes3_Det_ - DETERMINANTS NOTE A will denote a square...

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Unformatted text preview: DETERMINANTS NOTE: A will denote a square matrix [am] of size nxn. 1. DEFINITION: The (i,j) complement in A is a submatrix of A, denoted by A(i,j), which is obtained by deleting row i and column j from A. 2. DEFINITION: The determinant of A, denoted by det(A), is a scalar which is defined inductively as follows: 1. If n = l, det(A) = a11 If n = 2, det(A) = a a — a a 11 22 12 21 3. If n > 2, do the following: _ Define the (i,j) minor of A by Mij = det A(i,j) -Define the (i,j) cofactor of A by AH = (—l)i+j-Mij Then det(A) = a A + a A + a A + ... + a A 11 11 12. 12 13 13 1n in Note: The cofactor matrix is defined by cof(A) = [Aifl The adjoint matrix is adj(A) = cof(A)T EXAMPLE: A = 1 0 2 det(A) = 1.(_3) + 006 + 2-(—3) = -9 —l l 3 2 l 0 3. BIG THEOREM: a) For any row i : det(A) = a_A_ + a_A + a A + ... + a_A. 11 1 12 12 i3 13 1n 1n b) For any col 3 : det (A) = alelj + asz21+ a3jA3j+ . .. + anjAnj EXAMPLE: A = l O 2 COf (A) = —3 6 -3 -l l 3 2 —4 —l 2 l O -2 —5 1 row 1: det(A) = (l)(-3)+(O)(6)+(2)(-3) = -9 row 3: det(A) = (2)(—2)+(1)(-5)+(0)(1) = -9 col 2: det(A) = (O)(6)+(l)(-4)+(l)(-5) = -9 COFACTORS: A = 1 o 2 cof(A) = ’3 6 -3 adj(A) = —3 2 —2 —1 1 3 2 —4 -1 6 -4 -5 2 1 o —2 —5 1 -3 -1 1 A adj(A) = -9 o 0 A (—l/9)adj(A) = 1 0 0 o -9 0 o 1 0 o o -9 o 0 1 4. COROLLARY: l. A-adj(A) = det(A)-I n 2. If A is invertible, then A”'= (l/det(A))~adj(A). 5. DEFINITION: 1) A is lower triangular if all superdiagonal entries. are zero. 2) A is upper triangular if all subdiagonal entries are zero. 6. THEOREM: If A is (upper or lower) triangular, then det (A) = a11a22 nn 7. BIG THEOREM: det(AB) = det(A)det(B) 8. LEMMA: 1. If E = ri(i,j,I) , then det(E) = —l 2. If E = rm(r,i,I) , then det(E) = r 3. If E = ra(r,i,j,I) , then det(E) = l 9. THEOREM: 1. det(ri(i,j,A)) = —det(A) 2. det(rm(r,i,A)) = r°det(A) 3. det(ra(r,i,j,A) = det(A) 10. If A is tranformed to upper triangular form, Au, by t row operations, then A = E ...E E.A ‘x t 2 1 and det(Au) = det(Et)...det(E2)det(E1)det(A) 11. BIG THEOREM: Let A be an nxn matrix. The following is the procedure for the most efficient computation of det(A): 1) Reduce A to an upper triangular form, Au, using row operations. 2) Keep track of the operations, E1, and use the result above: If det(Et)...det(E2)det(E1) = then det(A) = (l/m)-det(Au) m, 12. EXAMPLE (Avoiding row interchange and row multiply) A=2—462 —>2—231——>2—231 —>2—231=A 5—963 01—9—2 01—9-2 01—9—2 t —12-6—2 00-3—1 00—3—1 00-3—1 2861 012 0-1 0010823 000—13 det(A) = (2)(1)(—3)(~13) =78 12. EXAMPLE: A: 1012 (1/4)(-2/11) 10 1 2 =A 3412‘_———“’ 01—1/2-1 “ 5100 00 1 2 0121 00 0—3 There are 7 row operations required to do the above reduction, but all have determinant = 1 (row adds) except the third (multiply row 2 by 1/4) and the sixth (multiply row 3 by -2/ll). Thus we need keep track of only the two row multiplies. det(Au) = det(E3) det(Eé) det(A). l-l-l-(—3) = (l/4)(—2/ll)-det(A) det(A) = (-3)(-22) = 66 ...
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