Question

Suppose CA=I. Show that the equation Ax=0 has only the trivial solution. Explain why A cannot have more columns than rows

Answer 1

um not entirely sure but i think that if there is more columns than rows then there could be a free variable that could take any value giving infinite solutions as for the first part of the question i dont have an answer for ya

Answer 2

Think of it this way, if A has more columns than rows that is equivalent to having more variables than equations. If you only have 3 equations and 4 unknowns then there is no way for you to solve the system without having a free variable.

Also, since you know CA=I then you know that A is an invertible matrix. Then, by definition, the column vectors of A must be linearly independent. If you have not learned about linear independence then maybe a more algebraic approach will make sense.

Since you know CA=I and therefore A is invertible, then you can do some algebra to the equation Ax=0 like so\[Ax=0\]\[A^{-1}Ax=A^{-1}0\]\[Ix=A^{-1}0\]\[x=0\]So as you can see, x equal to the zero vector is the only solution to that equation.
Hope this was helpful.