In a system of equations AX = B. by Cramer's rule if det(A)=0, then the system will have no solution but on the other hand, in non-homogeneous system of equations suppose the rank of square matrix nxn is (n-1) and rank of the augmented matrix[A|B] is also having the same rank but then we say that system is consistent, having rank < no. of variables implies infinitely many solutions.

how is this correct? I mean if the rank of A is n-1 < n and rank(A) = rank(A|B), then it also implies that determinant of square matrix nxn is zero but then we say it is consistent.

pls correct me if my theory is wrong too.

up vote 3 down vote accepted

Cramer rule doesn't says that when determinant is zero, it has no solution. When the determinant is zero, the system can either have no solution (inconsistent) or infinitely many solutions (consistent).

Cramer rule gives us a way to compute the solution when the solution is unique.

  • thanks , doing a question got mistaken somewhere – ashwani yadav Aug 10 at 5:19

For the system of two equations:

$ax+by=c$ and $Ax+By=C$

compute $D=\begin{vmatrix}a&b\\A&B\end{vmatrix}$, $D_x=\begin{vmatrix}c&b\\C&B\end{vmatrix}$ and $D_y=\begin{vmatrix}a&c\\A&C\end{vmatrix}$. Now you have two cases:

  1. If $D\ne 0$ then the unique solution is $x=\frac{D_x}{D}$ and $y=\frac{D_y}{D}$.
  2. If $D=0$ then Cramer's rule fails to give the solution (if it exists).
  • Since when could you divide matrices? – Chase Ryan Taylor Aug 10 at 6:12
  • Oh sorry! I used wrong brackets for determinats. – Mathlover Aug 10 at 6:13
  • $\ddot\smile{}{}{}{}{}{}{}{}{}{}$ – Chase Ryan Taylor Aug 10 at 6:14
  • Edited it @Chase Ryan Taylor. Thanks a lot! – Mathlover Aug 10 at 6:15

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