Question

Explain why the columns of an nxn matrix A are linearly independent when A is invertible Choose the correct answer below. O A. IFA is invertible, then for all x there is a b such that Ax=b. Since x = 0 is a solution of Ax0, the columns of A must be linearly independent OB. IA is invertible, then A has an inverse matrix A Since AA A AA must have linearly independent columns O C. If A is invertible, then A has an inverse matrix A . Since AA =I. A must have linearly independent columns. O D. ITA is invertible, then the equation Ax = 0 has the unique solution x = 0. Since Ax=0 has only the trivial solution, the columns of A must be linearly independent

Answer 1

Sol. eet let there Vig U2 ... Un denotes the columns of exist scalars 21,042 ...on such A , and that B This 21,41 +242 U2 + ... an Up = 0 can be written as Ax=o 1. 41=H2 = ... = H2=0 ( It ý true by defination of matrix multiblication Now show suppose that that the only i invertible. we want to solution to Ax= 0 x =0. multiplying both sides by A' A' CAX) = A'Co) cie. I x = 0 on 21 = 0 There fore the columns are einearly indebendent.