I would very much appreciate som explanations regarding trivial and non trivial solutions to a matrix (I am a beginner in studies of linear algebra).
Suppose that we have two matrices $A$ and $B$. The matrix $B$ is the row reduced form of matrix $A$ (by using a number of elementary row operations) and $B = A_k...A_2, A_1, A$, where $A_k...A_2, A_1$ are elementary matrices.
This means that the matrix $B$ is triangular and $det (B)$ is the product of the diagonal elements in the matrix $B$, the diagonal elements are either $1$ or $0$.
So far so good, Now the part which I don't really understand:
If any of the diagonal elements in $B$ is $0$ then the last row of $B$ must be $0$ (By shifting rows, so that the $0$ row is last?).
This means that the system $BX = 0$ has non trivial solutions (Why is that so? An explanation would be very much appreciated!).
This is also true for the equivalent system $AX=0$ and this means that $A$ is non invertible (An explanation how they make this conclusion would also be much appreciated).
Thank you kindly for your help, anything is helpful since I am trying to supplement my textbook to understand this better.