Linear Systems: Consistency & Cramer's Rule

For $AX = B$ (square): if $|A| \neq 0$ → unique solution. If $|A| = 0$: the system is inconsistent (no solution) or has infinitely many solutions, decided by whether the equations are genuinely contradictory or just dependent (e.g. two equations the same scaling but different constants → no solution). Three equations in TWO unknowns ($a_ix + b_iy + c_i = 0$, $i=1,2,3$) is a different shape: the system is consistent (all three lines pass through one point) iff $\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix} = 0$ — the same determinant that tests three lines for concurrency. Here a ZERO determinant is the consistent case, the opposite reading from the square system.

For $AX = B$ with $|A| = \Delta \neq 0$: $x = \Delta_x/\Delta$, $y = \Delta_y/\Delta$, $z = \Delta_z/\Delta$, where $\Delta_x$ is $\Delta$ with the $x$-column replaced by $B$ (and similarly for $\Delta_y, \Delta_z$).

$AX = 0$ has a non-trivial solution iff $|A| = 0$ (otherwise only $X = 0$). For $AX = B$: if $X_1 \neq X_2$ are both solutions, then $X_1 + t(X_2 - X_1)$ is a solution for every $t$ → infinitely many (the solution set is never exactly two).

Put the parameter into $|A|$ and solve $|A| = 0$ for the critical $k$. For each such $k$, substitute back: dependent-and-consistent → infinitely many; contradictory → no solution. Away from those $k$, the solution is unique.