# A Cramer's rule for $m \times n$ systems with unique solution?

I have a system of $n+1$ equations with $n$ unknowns for which I know the existence of a unique solution. I am interested in writing out an explicit formula for this solution (I am not concerned with speed of computation). For $n \times n$ systems, one could use Cramer's rule (http://mathworld.wolfram.com/CramersRule.html). Is there an analogous result for $m \times n$ systems with a unique solution?

• Are these all linear equations? If so, one of them is redundant and you can get rid of it – Cocopuffs Aug 24 '13 at 23:53
• crammers rule is useful when case has as many equations as number of unknowns , where you have a formula.Roughly speaking it is for square matrices . In other cases use rank method gaussian elimination to solve . The above is for $m \times n$ system , if you can reduce equations back to $n \times n$ then you can thereafter use crammers rule – Harish Kayarohanam Aug 25 '13 at 0:05

## 1 Answer

Yes. If you know the existence of a unique solution then what you can do is to calculate the determinant of submatrices until you find a nonzero one. In particular, select $n$ of the rows of your $m \times n$ matrix and compute the determinant. If it is nonzero then those rows are linearly independent and those $n$-equations suffice to determine the system. At that point, you can use Cramer's Rule on that sub-system.

All of this said, you could just use $\text{rref[ A|b]}$ and read off the answer just the same.