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Chapter 2 Section 2

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Lemma Let i=1 and j=2, then

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Lemma Let i=1 and j=2, then

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Lemma Let i=1 and j=2, then

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Lemma Let i=1 and j=2, then

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Lemma Let i=1 and j=2, then

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Lemma Let i=1 and j=2, then

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Lemma Notice where

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Switching Row 2 and Row 3, and calculating the determinant,

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Multiplying Row 3 by 4 and calculating the determinant,

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If E is an elementary matrix, then where If E is of type I If E is of type II If E is of type III

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Similar results hold for column operations. Indeed, if E is an elementary matrix, then E T is also an elementary matrix and

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I.Interchanging two rows (or columns) of a matrix changes the sign of the determinant. II.Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. III.Adding a multiple of one row (or column) to another does not change the value of the determinant

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I.Interchanging two rows (or columns) of a matrix changes the sign of the determinant. II.Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. III.Adding a multiple of one row (or column) to another does not change the value of the determinant

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I.Interchanging two rows (or columns) of a matrix changes the sign of the determinant. II.Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. III.Adding a multiple of one row (or column) to another does not change the value of the determinant

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I.Interchanging two rows (or columns) of a matrix changes the sign of the determinant. II.Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. III.Adding a multiple of one row (or column) to another does not change the value of the determinant

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I.Interchanging two rows (or columns) of a matrix changes the sign of the determinant. II.Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. III.Adding a multiple of one row (or column) to another does not change the value of the determinant

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I.Interchanging two rows (or columns) of a matrix changes the sign of the determinant. II.Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. III.Adding a multiple of one row (or column) to another does not change the value of the determinant

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