Given a quadratic form: $\displaystyle x^2 - 4xy + 6xz + 2xt + 4y^2 + 2yz + 4yt + 5z^2 - 6zt - t^2$, find the symmetric matrix that defines this, row reduce this matrix into row echelon form, and use this upper triangle matrix to complete the square and write the quadratic form as the sum/difference of squares.

So here is the matrix representation using x, y, z, and t as the diagonals from left to right and x being the first order, y second order, z third order, t fourth order:

$\displaystyle \left[ \begin{array}{cccc}1 & -2 & 3 & 1 \\-2 & 4 & 1 & 2 \\3 & 1 & 5 & -3 \\1 & 2 & -3 & -1 \\\end{array} \right]$

Row echelon:

$\displaystyle \left[ \begin{array}{cccc}1 & -2 & 3 & 1 \\0 & 7 & -4 & -6 \\0 & 0 & 7 & 4 \\0 & 0 & 0 & \frac{174}{49} \\\end{array} \right]$

NOTE: Rows 2 and 3 were switched during this process!! (It was unavoidable.)

Now, I'm not sure what to do with this matrix, as it's no longer the same matrix that represents the original quadratic.