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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.

Elimination goes directly to x. Elimination is also the way to calculate A 1,aswenow show. The Gauss-Jordan idea is to solve AA 1 D I, finding each column of A 1. A multiplies the first column of A 1 (call that x1/ to give the first column of I (call that e1/. This is our equation Ax1 D e1 D .1;0;0/. There will be two more equations.

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