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Chapter 2.2 Computing Determinants through RowReduction

# Proof write at bij where bij aji for any i n det

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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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