Finding the number of solutions of linear system Consider a system of m equations and n variables.
Where m≥n.
If homogeneous system AX=0 has only zero solution then how many solutions does corresponding non - homogeneous system AX=B have?
Since AX=0 has only zero solution so Det|A| ≠ 0. But for [A], m≠n. So determinant of A is of no use here. How do I sove this?
 A: Whilst expanding the answer, I decided to entirely rewrite it.
There are the following possibilities for a linear system of equations $Ax=b$ with an $m\times n$-matrix $A$:
1) $A$ is injective, i.e. $Ax=0$ implies $x=0$. In this case, a solution does not always exist but if it exists, it has to be unique. Equivalently, the rank of $A$ equals $n$.
2) $A$ is surjective, i.e. for all $b$ there is some $x$ such that $Ax=b$. In this case, a solution always exists but it may not be unique. Equivalently, the rank of $A$ equals $m$.
3) $A$ is bijective, i.e. satisfies 1) and 2). Thus there always exists precisely one solution. Equivalently, the rank of $A$ equals $m=n$.
4) $A$ is neither injective, nor surjective. There might be a solution or not and if you find one, it may be unique or not. In this case you are basically unable to make any general statement about the solutions.
Your question is settled in case 1). Thus, in general, there may be solutions or not. When does a solution exist? Precisely when $b$ is the column space of $A$, i.e. the vector space spanned by the columns of $A$. This is just the fact that the column space is the image (or range) of $A$. But whenever a solution exists, it has to be unique. Therefore, there can be none or one solution.
If you additionally know that $m=n$, then there is always precisely one solution.
If something remained unclear, don't hesitate to ask.
