In previous sections, we solved linear systems using Gauss elimination method or the Gauss-Jordan method. In the examples considered, we have encountered three possibilities, namely
Based on the above possibilities, we have the following definition.
The question arises, as to whether there are conditions under which the linear system is consistent. The answer to this question is in the affirmative. To proceed further, we need a few definitions and remarks.
Recall that the row reduced echelon form of a matrix is unique and therefore, the number of non-zero rows is a unique number. Also, note that the number of non-zero rows in either the row reduced form or the row reduced echelon form of a matrix are same.
The reader is advised to supply a proof.
Therefore, we can define column-rank of as the number of non-zero columns in It will be proved later that
Thus we are led to the following definition.
We now apply column operations to the matrix Let be the matrix obtained from by successively interchanging the and column of for Then the matrix can be written in the form where is a matrix of appropriate size. As the block of is an identity matrix, the block can be made the zero matrix by application of column operations to This gives the required result. height6pt width 6pt depth 0pt
as the elementary martices 's are being multiplied on the left of the matrix Let be the columns of the matrix . Then check that for . Hence, we can use the 's which are non-zero (Use Exercise 18.104.22.168) to generate infinite number of solutions. height6pt width 6pt depth 0pt
If is invertible and then show that
A K Lal 2007-09-12