I don't quite understand how to use that augmented matrix to prove anything about a, b and c.
Any ideas on where to start?
25. You know four points that lie on the curve, $\displaystyle \displaystyle (0, 10), (1, 7), (3, -11), (4, -14)$.
So that means they all satisfy the equation $\displaystyle \displaystyle y = a\,x^3 + b\,x^2 + c\,x + d$.
So
$\displaystyle \displaystyle 10 = a\cdot 0^3 + b\cdot 0^2 + c\cdot 0 + d$
$\displaystyle \displaystyle 7 = a\cdot 1^3 + b\cdot 1^2 + c\cdot 1 + d$
$\displaystyle \displaystyle -11 = a\cdot 3^3 + b\cdot 3^2 + c\cdot 3 + d$
$\displaystyle \displaystyle -14 = a\cdot 4^3 + b\cdot 4^2 + c\cdot 4 + d$.
So now you can set up your matrix equation...
$\displaystyle \displaystyle \left[\begin{matrix}0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1\\ 27 & 9 & 3 & 1 \\ 64 & 16 & 4 & 1\end{matrix}\right]\left[\begin{matrix}a \\ b\\ c\\d \end{matrix}\right] = \left[\begin{matrix}\phantom{-}10 \\ \phantom{-}7 \\ -11 \\ -14\end{matrix}\right]$
.I usually write the constants in the same brackets as the large matrix here (to the furthest right side of the matrix, since that's what we were taught). So:
0 0 0 1 10
1 1 1 1 7
27 9 3 1 -11
64 16 4 1 -14
Is that right?
Would I use Gauss-Jordan to reduce that into reduced row-echelon?
I'm still confused about how to find the variables from there.
[QUOTE=TN17;604092].I usually write the constants in the same brackets as the large matrix here (to the furthest right side of the matrix, since that's what we were taught). So:
0 0 0 1 10
1 1 1 1 7
27 9 3 1 -11
64 16 4 1 -14
Is that right?/[quote]
Yes, that is what is meant by the "augmented" matrix.
Yes, you certainly could use Gauss-Jordan. I would be inclined to swap the first row to the last initially to getWould I use Gauss-Jordan to reduce that into reduced row-echelon?
I'm still confused about how to find the variables from there.
$\displaystyle \begin{bmatrix}1 & 1 & 1 & 1 & 7 \\ 27 & 9 & 3 & 1 & -11\\ 64 & 16 & 4 & 1 & -14 \\ 0 & 0 & 0 & 1 & -14\end{bmatrix}$
Then subtract 27 times the first row from the second, 64 times the first row from the third, etc.