Transcription of 2.5 Inverse Matrices - MIT Mathematics
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Example: air traffic controller
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).
illustrates a basic rule of mathematics: Inverses come in reverse order. It is also common sense: If you put on socks and then shoes, the first to be taken off are the . The same reverse order applies to three or more matrices: Reverse order.ABC/ 1 D C 1B 1A 1: (5) Example 2 Inverse of an eliminationmatrix.IfE subtracts 5 times row 1 from row 2,
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