Question

Assume that the matrices A and B are square and the same size. Decide if each of the following is true or false, and you must provide a proof as to why it is either true or false. Please be clear and write with understandable handwriting so that I am able to understand it.

☺ If Až=ŏ has non-trivial solutions, then AT - has linemly dependent columns, © If The row vectors of A are dependent, then 1. so are the column vectas : ® If there is a } for which Ax=] is inconsistent, then Ar=b is inconsistent for all chuicer o B.

Answer 1

ca AX=0 has a non- trivial solution. This means that if AX=0 has an equations and m unknowns then m>nj.e. number of variables are more than equations. This implies that are there must be a free variable and therefore, columns of A are linearly dependent. This proves the statement. (b) To prove this statement, we transform. the mateix to a echelon form" using elementary operations. This will not change number of L. I. rows 4 columns. In that matrix No. of independent rows will equal to no. of independent columns obviously. Hence proved (c) This is false. For example- 727 : 7 15 or AX =B op 31 or Ax=B AA & rank [AB] This is inconsistent as rank but if we take B=557 then 2 [3) 13-15 per in or will be consistent because rank A = rank [AB] =).