Example: tourism industry

2.5 Inverse Matrices

Inverse Inverse Matrices '&$%1If the square matrixAhas an Inverse , then bothA 1A=IandAA 1= test invertibility is elimination :Amust haven(nonzero) for invertibility is the determinant ofA:detAmust not be tests for invertibility isAx=0:x=0must be the only (same size) are invertible then so isAB:|(AB) 1=B 1A 1=Iisnequations forncolumns ofA 1. Gauss-Jordan eliminates[A I]to[I A 1].7 The last page of the book gives14equivalent conditions for a squareAto be a square matrix. We look for an Inverse matrix A 1of the same size,such thatA 1timesAequalsI. WhateverAdoes,A 1undoes. Their product is theidentity matrix which does nothing to a vector, soA 1Ax= 1might not a matrix mostly does is to multiply a vectorx.

Multiply EE−1 to get the identity matrix I. Also multiply E−1E to get I. We are adding and subtracting the same 5 times row 1. If AC = I then automatically CA = I. For square matrices, an inverse on one side is automatically an inverse on the other side. Example 3 Suppose F subtracts 4 times row 2 from row 3, and F−1 adds it back: F =

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