Lecture17 - University of Toronto at Scarborough Department...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 17 Section 4.1, Determinants Sophie Chrysostomou Areas, Volumes and Cross Products definition 0.1 (i) The determinant of 1 × 1 matrix is its sole entry. (ii)The determinant of a 2 × 2 matrix is given by det(A) = a1 a2 b1 b2 = a1 b2 − b1 a2 , where A = a1 a2 b1 b2 (iii) The determinant of a 3 × 3 matrix is given by det(A) = a1 a2 a3 b1 b2 b3 c1 c2 c3 = a1 b2 b3 c2 c3 − a2 b1 b3 c1 c3 + a3 b1 b2 c1 c2 definition 0.2 If a = [a1 , a2 , a3 ], b = [b1 , b2 , b3 ] ∈ R3 the cross product of a and b is given by a×b = a2 a3 aa aa ,− 1 3 , 1 2 b2 b3 b1 b3 b1 b2 =i a2 a3 aa aa −j 1 3 +k 1 2 b2 b3 b1 b3 b1 b2 Note: a × b is perpendicular to both a and b. (Show this on your own.) 1 theorem 0.3 (i) If a parallelogram is determined by two nonzero vectors, a = [a1 , a2 ] and b = [b1 , b2 ] in R2 , then its area is given by Area = |a1 b2 − a2 b1 | = det a1 a2 b1 b2 = a1 a2 b1 b2 (ii) If a parallelogram is determined by two nonzero vectors, a = [a1 , a2 , a3 ] and b = [b1 , b2 , b3 ] in R3 , then its area is given by a × b . (iii) If a parallelepiped is determined by three nonzero vectors a = [a1 , a2 , a3 ], b = [b1 , b2 , b3 ] and c = [c1 , c2 , c3 ] in R3 , then the volume of the box is given by V olume = |a · (b × c)| 2 The Determinant of a Square Matrix definition 0.4 The determinant of 1 × 1, 2 × 2 and 3 × 3 matrices are defined. Let n > 1 and suppose that the determinant of (n − 1) × (n − 1) matrices is defined. Let A = [aij ] be an n × n matrix. 1. The minor matrix Aij is the (n − 1) × (n − 1) matrix obtained by removing the ith row and j th column of A. 2. The cofactor of aij of A is aij = (−1)i+j det(Aij ) = (−1)i+j |Aij | 3. The determinant of A is det(A) = a11 a12 · · · a1n a21 a22 · · · a2n . . . . . . . . . an1 an2 · · · ann = a11 a11 + a12 a12 + · · · + a1n a1n n = n (−1)(i+1) a1i |A1i | i=1 expansion along j th row i=1 n expansion along first row (−1)(i+j ) aji |Aji | a1 i a1 i = n = aji aj i = i=1 i=1 n n = (−1)(i+j ) aij |Aij | expansion along j th column aij aij = i=1 i=1 3 theorem 0.5 Let A, C be n × n matrices. Then: 1. det(A) = det(AT ). 2. A Ri ←→ Rj B , then det(B ) = −det(A) 3. A Ri −→ rRi B , then det(B ) = r det(A) 4. A Ri −→ Ri + rRj B , then det(B ) = det(A) 5. If A contains proportional rows (or columns), then det(A) = 0. 6. A is invertible 7. det(AB ) = det(A)det(B ). 8. If A is invertible, then det(A−1 ) = 9. ⇐⇒ det(A) = 0. 1 . det(A) If A is a triangular matrix, then det(A) is the product of all of its entries along the main diagonal. 4 ...
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This note was uploaded on 05/22/2010 for the course MATH MATA23 taught by Professor Sophie during the Spring '10 term at University of Toronto- Toronto.

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