System of linear equations with a unique solution
Suppose the system of linear equations $AX=B$ has a unique solution for some $B$ . Prove that rref of $A$ is the same as $I_n$. ($A$ is a square matrix)
My try : Because the system has a unique solution therefore it's possible to write $x_1 = a , x_2 = b , \dots , x_n = z$ . In the matrix form, for the coefficient matrix, we have :
\begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}I don't know whether that's enough for proving that statement or not .
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$\begingroup$If there is a unique solution, then there must be n pivots, or else there will be other variables defined uniquely by a pivot which leads to a n-dimensional subspace of pivots. That is, a unique solution only exists when the rank is maximized.
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