Chapter 2.3 Determinants-Properties

Outline linearity of the determinant determinants and

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Unformatted text preview: the Determinant Determinants and Invertibility Determinants and Products Determinants and Inverses The Determinant and Invertibility Theorem An n × n matrix A is invertible if and only if det A = 0. Proof. To obtain rref (A), swap rows s times and divide various rows by the nonzero scalars λ1 , λ2 , . . . , λr Then det A = (−1)s λ1 λ2 · · · λr det(rref (A)). A invertible: rref (A) = In and det In = 1, so det A = (−1)s λ1 λ2 · · · λr = 0. A not invertible: The last row of rref (A) is a row of zeros, so det(rref (A)) = 0 and therefore det A = 0. Outline Linearity of the Determinant Determinants and Invertibility Determinants and Products Determinants and Inverses The Invertibility Theorem Theorem Suppose A is an n × n ma...
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