16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-16Thus, we can form a big augmented matrix of the formA | I.This is an n × 2n matrix (with the vertical line in the middle).Applying the row operation method for solving linear systems to thismatrix, we do a sequence of successive row modifications. If A is actuallyinvertible, then, at the end of doing these operations, we will actually get ann × 2n matrix of the formI | B.It turns out that the resulting matrix B is actually A −1 .The proof of this is not difficult. It amounts to the following observation.Let us use the notation op to denote any of the elementary row operations.Write A op B to mean that B is obtained from A by applying the elementaryrow operation op.As above, let I be the n × n identity matrix.Then the following is true.Proposition Let I denote that n × n identity matrix. Let op be anyelementary row operation, and assume that A op B and I op D.Then,B = DAThis means that we get B from A doing a left multiplication by D whereD is gotten from I via the same elementary row operation used to get B fromA.Let us describe this in more detail. It will be convenient to have specificnotations for the matrices gotten by applying elementary row operations toI.There are of three types:Let i, j be any two integers with 1 ≤ i ≤ n and 1 ≤ j ≤ n.1. Interchange of rowsLet P ij denote the matrix obtained by interchanging the i−th and j−throws of I. (This is usually called a permutation matrix since it permutesthe rows of I).