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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
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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
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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.
 Spring '10
 Sophie
 Linear Algebra, Algebra, Determinant

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