L05M - Determinants Consider the two linear equations for...

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Unformatted text preview: Determinants Consider the two linear equations for two unknowns x, y xa1 + ya1 = b1 1 2 xa2 + ya2 = b2 1 2 Do xa1a2 + ya1a2 = b1a2 12 22 2 xa2a1 + ya2a1 = b2a1 12 22 2 and subtract: x(a1a2 − a2a1) = b1a2 − b2a1 . 12 12 2 2 Do xa1a2 + ya1a2 = b1a2 11 21 1 xa2a1 + ya2a1 = b2a1 11 21 1 ( ∗x ) ( ∗) and subtract: (∗y ) y (a1a2 − a2a1) = b2a1 − b1a2 . 12 12 1 1 We see that (∗x), (∗y ) have a solution no matter what b1, b2 are if and only if the determinant a1 a1 def 1 2 1 2 = (a1a2 − a2a1) 12 a2 a2 1 2 of the Coefficient Matrix a1 a1 1 2 a2 a2 1 2 of (∗) is not zero. In that case we get b1a2−b2a1 x= 12 22 a1a2−a1a1 2 2 b2a1−b1a2 y= 11 21 a1a2−a1a1 2 2 2 ...
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L05M - Determinants Consider the two linear equations for...

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