proof of Cramer’s rule
We claim that this implies that the equation has a unique solution. Note that is a solution since , so we know that a solution exists.
Let be an arbitrary solution to the equation, so . But then , so we see that is the only solution.
We know that for any matrices that the th column of the product is simply the product of and the th column of . Also observe that for . Thus, by multiplication, we have:
Since is with column replaced with , computing the determinant of with cofactor expansion gives:
Thus by the multiplicative property of the determinant,
and so as required.
|Title||proof of Cramer’s rule|
|Date of creation||2013-03-22 13:03:24|
|Last modified on||2013-03-22 13:03:24|
|Last modified by||rmilson (146)|