Lecture Slides 12

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

{[ snackBarMessage ]}

Page1 / 13

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online