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2.5 Inverse Matrices - MIT Mathematics

Inverse Inverse MatricesSupposeAis a square matrix . We look for an Inverse matrix A 1of the same size, suchthatA 1timesAequalsI. WhateverAdoes,A 1undoes. Their product is the identitymatrix which does nothing to a vector, soA 1might not a matrix mostly does is to multiply a vectorx. MultiplyingAxDbbyA 1givesA 1 AxDA isxDA 1b. The productA 1 Ais like multiplying bya number and then dividing by that number. A number has an Inverse if it is not zero Matrices are more complicated and more interesting. The matrixA 1is called Ainverse. DEFINITIONThe matrixAisinvertibleif there exists a matrixA 1such thatA 1 ADIandAA 1DI:(1)Not all Matrices have inverses. This is the first question we ask about a square matrix :IsAinvertible? We don t mean that we immediately calculateA 1. In most problemswe never compute it! Here are six notes aboutA 1 The Inverse exists if and only if elimination producesnpivots(row exchangesare allowed).

2.5. Inverse Matrices 81 2.5 Inverse Matrices Suppose A is a square matrix. We look for an “inverse matrix” A 1 of the same size, such that A 1 times A equals I. Whatever A does, A 1 undoes. Their product is the identity matrix—which does nothing to a vector, so A 1Ax D x. But A 1 might not exist. What a matrix mostly does is to multiply ...

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