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Unformatted text preview: then det AT = det A.
Proof.
Write AT = [bij ], where bij = aji . For any i ,
n det A = n aki Cki = det AT . aik Cik =
k =1 k =1 Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix Triangular Matrices
Theorem
If A = [aij ] is an n × n triangular matrix, then
det A = a11 a22 · · · ann
Proof.
Prove ﬁrst for lower triangular matrices using induction.
Example A= 3
2
3
7
0 −1
2 −4
0
0 −3 −6
0
0
0
6
0
0
0
0 −8
−4
−2
7
−2 Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix The Determinant and Row Operations
Theorem
Let A be a square matrix.
1 If B is obtained from A by interchanging two rows (or two
columns) of A, then det B = − det A. 2 If B is obtained from A by multiplying a row (or column) of A
by a scalar k (k = 0), then det B = k det A. 3 If B is obtained from A by adding a scalar multiple of one row
(resp., column) to another row (resp...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.
 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Determinant, Matrices

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