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Unformatted text preview: then det AT = det A.
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
If A = [aij ] is an n × n triangular matrix, then
det A = a11 a22 · · · ann
Prove ﬁrst for lower triangular matrices using induction.
Example A= 3
0 −3 −6
−2 Outline Determinants of Special Matrices Determinants Via Elementary Row Operations Determinant of an Elementary Matrix The Determinant and Row Operations
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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