Lecture Slides 12

# Lecture Slides 12 - AMS 210 Applied Linear Algebra AMS 210...

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Unformatted text preview: AMS 210: Applied Linear Algebra October 20, 2009 AMS 210 Topics Today Matrix Inverses Computing Matrix Inverses Use of Inverse in Multiple Right-Hand Sides Reversing a Markov Chain Properties of the Inverse Fundamental Theorem of Solving Ax = b . AMS 210 Matrix Inverses and Solutions to Linear Systems Additive inverse: A + (- A ) = 0 Multiplicative inverse: AA- 1 = I and A- 1 A = I Theorem 1: If A has an inverse A- 1 , then the system of equations Ax = b has the solution x = A- 1 b (proof) A matrix is invertible (nonsingular) if it has an inverse. Singular matrices have no inverse. Inverses are unique. AMS 210 Matrices With and Without Inverses 1 Matrix A = 3 1 4 2 has the inverse A- 1 = 1- 1 2- 2 3 2 . 2 Show that both AA- 1 = I and A- 1 A = I . 3 A matrix with rows that are multiples of other rows have no inverses. Reductio: suppose an inverse C to such a matrix B existed. Then BC = b R 1 · c C 1 b R 1 · c C 2 b R 2 · c C 1 b R 2 · c C 2 = I . But the second row of BC both must be a multiple of the first row and cannot be, because that’s not so in I . QED (for the 2x2 case anyway. Its generalizations should be fairly clear)....
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Lecture Slides 12 - AMS 210 Applied Linear Algebra AMS 210...

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