There are many ways to prove this rule.
Here is an intuitive argument for it.
If we were in one dimension then our one equation would read Ax = B and the solution would be . In more dimensions we want to eliminate the effects of the other dimensions, and distil out of our several equations an equation like this one dimensional one.
Suppose we were in three dimensions, for example. Then the determinant with the given x, y and z columns is then the signed volume of the figure whose x and y dependence is determined by the y and z columns. This volume is the "height" in the x direction multiplied by some measure of the figure in the y and z directions.
If we substitute the right hand side for the x column in this determinant, we get the volume of the figure that is identical in the y and z directions, but has a "height" in the x direction determined by the right hand side.
The ratio of the two determinants will therefore be (assuming they are not both 0) the ratios of the two heights: in the x direction, which is exactly what we are looking for! So x is the ratio of the two. (Obviously, this argument applies similarly in all dimensions and to any component.) And Cramer's rule is proven.
You call that a proof? That's way too vague.
OK, OK, here is another stab at it.
Suppose the elements in the first (or any other) row in the y and z columns are both 0. Then the corresponding equation has no y or z term and gives the answer we want. We need look no further.
If we look at the determinant equation implied by Cramer's rule in this case, the arrays are partially row reduced, and we can read off the same answer, with both terms multiplied by some factor from the other rows and the y and z columns. So the result is proven in this case.
And what if the arrays lack this property?
You can subtract multiples of the other rows from the one we have been talking about, and row reduce so that the entries in the y and z column both become 0, and do the identical thing for the both arrays in Cramer's rule simultaneously.
When you are done, we know the rule is correct, by the first claim. And the row reductions did not change either determinant at all. So the rule must have been correct from the start!
Exercise: Convince someone else, anyone, that this Cramer's rule is
correct by explaining a proof of it to that person, (either one of the proofs
above or any other.)
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