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3.3 Properties of Determinants Study Book 3.3 & Larson , Ch 3.3

3.3 Properties of Determinants Study Book 3.3 & Larson , Ch 3.3. Objectives: know that det ( AB ) = det ( A ) det ( B ) det ( A -1 ) = 1 / det( A ) det ( A T ) = det ( A ) det ( cA ) = c n det ( A ) ( c scalar, A n x n )

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3.3 Properties of Determinants Study Book 3.3 & Larson , Ch 3.3

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  1. 3.3 Properties of DeterminantsStudy Book 3.3 & Larson, Ch 3.3 Objectives: know that • det (AB) = det (A) det (B) • det (A-1) = 1 / det(A) • det (AT ) = det (A) • det (cA) = cn det (A) (c scalar, Anxn) • A is invertible if and only if det(A)  0 • Ax=b has a unique solution for every nx1 matrix b, if & only if det(A) is not 0.

  2. Th 3.5 uses elementary matrices to prove det(AB) = det(A) det(B). Now if A is invertible,A A-1 = I det (A A-1) = det (I) det (A) det (A-1) = det ( I ) (Th above) det (A) det (A-1) = 1 (diagonal product) Since the RHS is 1, neither of the dets on the LHS can be 0, so we can divide: det(A) = 1/ det(A-1)& vice versa. Summary: if A is invertible, det(A)  0 * and det (A-1) = 1 / det (A)

  3. If A is not invertible, then A does not row-reduce to I. ie row-reducingyields a row of 0’s. ie the reduced form of A has determinant 0. Since row ops can only change a det by a non-zero factor. A must also have determinant 0. Hence if A is not invertible,det (A) = 0. ** Results * & ** yield A is invertible iff det(A)  0. Also, since a determinant can be expanded by anyrow or column Det (AT) = Det (A) .

  4. Lastly, We already know that for any nx1matrix b (RHS’s) Ax = b has a unique solution iff A is invertible. It follows that Ax = b has a unique solutioniff det(A) is not 0. ________________________________________________________________________________ Recall: Multiplying a row of A by a constant c multiplies the determinant by c. ie, if A is nxn, det (cA) = cn det(A) It looks as though a factor can be “factored out” of onerow at a time (or column). ___________________________________________________________________________________ Example: If A is a 3x3 matrix, and det (A) = 5, (a) det (AT) = 5 (b) det(2A) = 8 x 5 = 40 (3 rows with factor 2) (c) det (2 A-1) = 8 det (A-1) = 8 (1/5) = 8/5.

  5. Homework Larson & Edwards Ch 3.3 • Master 1 - 34. • Write full solutions to Q 7, 9, 19, 23, 25, 29, 31, and Ed 4, Q 47, 48 or Ed 5, Q 49, 50.

  6. Larson 3.4 is optional: Browse for interest!________________________________3.4 Summary & ReviewStudy Book p 48. • See summaries on determinants: • see Larson or last 2 pages of Study Guide App A. • Try some Ch 3 Review exercises (but remember we omitted topics) • Note more applications of determinants in the Ch 3 Projects. • Try some of the Ch 1-3 Cumulative Test.

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