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Unformatted text preview: trix. Then the following are equivalent
(they are all true or they are all false).
(a) A is invertible.
(b) Ax = 0 has only the trivial solution.
(c) rref (A) = In .
(d) A = E1 E2 . . . Er for some elementary matrices E1 , E2 , . . . , Er .
(e) Ax = b has exactly one solution for every n × 1 column vector
(f) det A = 0. Outline Linearity of the Determinant Determinants and Invertibility Determinants and Products Determinants and Inverses The Determinant of a Product with an Elementary Matrix
If A is an n × n matrix and E is an n × n elementary matrix, then
det(EA) = (det E )(de...
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