Consider the matrix A = [[1,a,a^2],[1,b,b^2],[1,c,c^2]] Assume that a, b, c are different numbers. Consider points P(a, y_1), Q(b, y_2), R(c, y_3). Set up the system to find a parabola y = d + ex +...

Consider the matrix

A = [[1,a,a^2],[1,b,b^2],[1,c,c^2]]

Assume that a, b, c are different numbers. Consider points P(a, y_1), Q(b, y_2), R(c, y_3).

Set up the system to find a parabola

y = d + ex + fx_2 passing by P,Q,R.

What are the values of d, e and f in terms of a,b and c?

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We can write the system as

`A([d],[e],[f]) = ([y_1],[y_2],[y_3])`

and can solve it by noting that

`A^(-1)([y_1],[y_2],[y_3]) = ([d],[e],[f])`

We find the inverse of the matrix A by using the formula

`A^(-1) = 1/det(A) C^(T)`

where `C^T` is the cofactor matrix of A. The (i,j) entry of C is (-1)^(i+j) times the (i,j) minor of A, obtained by taking the determinant of the matrix arrived at when deleting the ith row and jth column of A.

Using row reduction

`A = ([1,a,a^2],[0,b-a,b^2-a^2],[0,c-a,c^2-a^2])`

Then,

`C = ([+|[b-a,b^2-a^2],[c-a,c^2-a^2]| - |[0,b^2-a^2],[0,c^2-a^2]| + |[0,b-a],[0,c-a]|],[-|[a,a^2],[c-a,c^2-a^2]| + |[1,a^2],[0,c^2-a^2]| + |[1,a],[0,c-a]|],[+|[a,a^2],[b-a,b^2-a^2]| - |[1,a^2],[0,b^2-a^2]| + |[1,a],[0,b-a]|])`

`= ([(b-a)(c-a)(c-b),0,0],[-ac(c-a),(c+a)(c-a),(c-a)],[ab(b-a),a^2-b^2,b-a])`

So that

`C^T =([(b-a)(c-a)(c-b),-ac(c-a),ab(b-a)],[0,(c+a)(c-a),a^2-b^2],[0,c-a,b-a])`

Now, since det(A) = (b-a)(c-a)(c-b) we have that

`A^(-1) = C^T/((b-a)(c-a)(c-b))`

`= ([1,-(ac)/((b-a)(c-b)),(ab)/((c-a)(c-b))],[0,(c+a)/((b-a)(c-b)),-(a+b)/((c-a)(c-b))],[0,1/((b-a)(c-b)),1/((c-a)(c-b))])`

Finally then, we have that

`([d],[e],[f]) = A^(-1)([y_1],[y_2],[y_3])`

where `A^(-1)` is as given above.

For example `d = y_1 -((ac)/((b-a)(c-b)))y_2 + ((ab)/((c-a)(c-b)))y_3`