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Lecture 6 Intersection of Hyperplanes and Matrix Inverse Shang-Hua Teng

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Expressing Elimination by Matrix Multiplication

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Elementary or Elimination Matrix The elementary or elimination matrix That subtracts a multiple l of row j from row i can be obtained from the identity matrix I by adding (-l) in the i,j position

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Elementary or Elimination Matrix

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Pivot 1: The elimination of column 1 Elimination matrix

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The Product of Elimination Matrices

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Elimination by Matrix Multiplication

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Linear Systems in Higher Dimensions

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Booking with Elimination Matrices

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Multiplying Elimination Matrices

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Inverse Matrices In 1 dimension

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Inverse Matrices In high dimensions

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Inverse Matrices In 1 dimension In higher dimensions

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Some Special Matrices and Their Inverses

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Inverses in Two Dimensions Proof:

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Uniqueness of Inverse Matrices

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Inverse and Linear System

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Therefore, the inverse of A exists if and only if elimination produces n non-zero pivots (row exchanges allowed)

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Inverse, Singular Matrix and Degeneracy Suppose there is a nonzero vector x such that Ax = 0 [column vectors of A co-linear] then A cannot have an inverse Contradiction: So if A is invertible, then Ax =0 can only have the zero solution x=0

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One More Property Proof So

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Gauss-Jordan Elimination for Computing A -1 1D 2D

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Gauss-Jordan Elimination for Computing A -1 3D

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Gauss-Jordan Elimination for Computing A -1 3D: Solving three linear equations defined by A simultaneously n dimensions: Solving n linear equations defined by A simultaneously

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Example:Gauss-Jordan Elimination for Computing A -1 Make a Big Augmented Matrix

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Example:Gauss-Jordan Elimination for Computing A -1

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