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Lecturenotes 31january2011

# Lecturenotes 31january2011 - Section 2.2 Evaluating...

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Section 2.2: Evaluating Determinants by Row Reduction We will look at the relationship between the determinants of row-equivalent matrices. Recall we have 3 elementary matrix operations 1. Interchange two rows 2. Muptiply a row bya nonzero constant 3. Add a multiple of one row to another row Let’s see the effects of these operations on a 2 × 2 matrix. Let A = a b c d (a) A = a b c d R 1 R 2 B = c d a b det A = ad - bc, det B = cb - ad = - ( ad - bc ) Observation: Interchanging two rows effected the determinat by a (-1) factor. That is det ( B ) = - det ( A ) (b) A = a b c d 2 R 1 B = 2 a 2 b c d det A = ad - bc, det B = 2 ad - 2 bc = 2( ad - bc ) Observation: Multiplying a row by 2 (any number k) effected the determinat by a factor 2 (k). That is det ( B ) = 2 det ( A ) (in general det ( B ) = k det ( A ) ) (c) A = a b c d 2 R 1 + R 2 B = a b c + 2 a d + 2 b det A = ad - bc, det B = a ( d + 2 b ) - b ( c + 2 a ) = ( ad - bc ) Observation: Adding a multiple of a row to another row does not change the determinat. That is det ( B ) = det ( A ) These observations in fact apply to any size square matrix.

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