How do I prove Cramer's Rule?
[Example of Cramer's]
ax+by=e,
cx+dy=f
Cramer's Rule states that if \(\begin{bmatrix} a && b \\ c && d \end{bmatrix}\) is nonzero, then the solution to the system [left above] is x = \(\begin{bmatrix} e && b \\ f && d \end{bmatrix} \over \begin{bmatrix} a && b\\ c && d \end{bmatrix}\) and y = \(\begin{bmatrix} a && e \\ c && f \end{bmatrix} \over \begin{bmatrix} a && b \\ c && d \end{bmatrix}\)
How Cramer's Rule works: \(\begin{bmatrix} a && b \\ c && d \end{bmatrix}\) = ad - bc
+23246 ax + by = e ---> · d ---> adx + bdy = de
cx + dy = f ---> · -b ---> -bcx - bdy = -bf
---> (ad - bc)x = (de - bf)
---> x = (de - bf) / (ad - bc)
ax + by = e ---> ·-c ---> -acx - bcy = -ce
cx + dy = f ---> · a ---> acx + ady = af
---> (ad - bc)y = (af - ce)
---> x = (af - ce) / (ad - bc)
which match the solutions that you get when you use Cramer's formula.
