Question

Matrix conception question please able to follow the comment

1. If the nxn matrix has a unique solution, that is invertible. true or false

2. all invertible matrix only has one solution true or false and explain

Answer 1

$Q1.~Let~A~be~a~matrix~of~order~n\times~n.~Consider~system~of~non-homogeneous~equation:\\ Ax=b~where~x~is~unknown~vector~and~b~is~known~vector~and~assume~it~is~consistent.\\ Now,~let~there~exist~a~matrix~B~such`that~BA=I_n=identity~matrix~of~\\ order~n\\ Then~BAx=BA~or,~x=Bb~so~x~has~a~unique~solution~iff~B~is~the~inverse\\ of~A~since~B~is~unique.\\ Hence~if~the~n\times~n~matrix~has~a~unique~solution~then~the~matrix~is~invertable.\\ Hence~the~statement~is~true.\\ Q2.~From~above~we~see~that~Ax=b~has~only~one~solution~iff~A~is~invertable.\\ Hence~the~statement~is~true.$