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# Lecture17 - University of Toronto at Scarborough Department...

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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 ) = a 1 a 2 b 1 b 2 = a 1 b 2 - b 1 a 2 , where A = a 1 a 2 b 1 b 2 (iii) The determinant of a 3 × 3 matrix is given by det ( A ) = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 b 3 c 2 c 3 - a 2 b 1 b 3 c 1 c 3 + a 3 b 1 b 2 c 1 c 2 definition 0.2 If a = [ a 1 , a 2 , a 3 ] , b = [ b 1 , b 2 , b 3 ] R 3 the cross product of a and b is given by a × b = a 2 a 3 b 2 b 3 , - a 1 a 3 b 1 b 3 , a 1 a 2 b 1 b 2 = i a 2 a 3 b 2 b 3 - j a 1 a 3 b 1 b 3 + k a 1 a 2 b 1 b 2 Note: a × b is perpendicular to both a and b . (Show this on your own.) 1

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theorem 0.3 (i) If a parallelogram is determined by two nonzero vectors, a = [ a 1 , a 2 ] and b = [ b 1 , b 2 ] in R 2 , then its area is given by Area = | a 1 b 2 - a 2 b 1 | = det a 1 a 2 b 1 b 2 = a 1 a 2 b 1 b 2
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Lecture17 - University of Toronto at Scarborough Department...

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