Determinant equation???

[Guest]

Have no idea what to do here. Any suggestions? The task is to solve the determinant equation.


[tex]\begin{pmatrix} 1&1&1&...&1\\1&1-x&1&...&1\\1&1&2-x&...&1\\...&...&...&...&...\\1&1&1&...&n-x \end{pmatrix} = 0[/tex]

[Guest]

Subtracting one row from another (distinct) row doesn't affect the determinant. So we can subtract the first row from all the other rows to get
[tex]\begin{vmatrix}
1 & 1 & 1 & \ldots & 1\\
1 & 1-x & 1 & \ldots & 1\\
1 & 1 & 2-x & \ldots & 1\\
\ldots & \ldots & \ldots & \ldots & \ldots\\
1 & 1 & 1 & \ldots & n-x
\end{vmatrix}
=
\begin{vmatrix}
1 & 1 & 1 & \ldots & 1\\
0 & -x & 0 & \ldots & 0\\
0 & 0 & 1-x & \ldots & 0\\
\ldots & \ldots & \ldots & \ldots & \ldots\\
0 & 0 & 0 & \ldots & n-1-x
\end{vmatrix}[/tex]

We can also subtract columns without affecting the determinant, in particular we can subtract the first column from all other columns to get
[tex]\begin{vmatrix}
1 & 0 & 0 & \ldots & 0\\
0 & -x & 0 & \ldots & 0\\
0 & 0 & 1-x & \ldots & 0\\
\ldots & \ldots & \ldots & \ldots & \ldots\\
0 & 0 & 0 & \ldots & n-1-x
\end{vmatrix}[/tex]

The determinant of a diagonal matrix is just the product of it's diagonal elements, so the determinant is
[tex]-x(1-x)(2-x)\ldots(n-1-x)[/tex]
Which is equal to [tex]0[/tex] when [tex]x=0, 1, 2, \ldots, n-1[/tex].

Another way to solve this would be to realize that the determinant will be a degree [tex]n[/tex] polynomial in [tex]x[/tex], so there will be [tex]n[/tex] roots or values of [tex]x[/tex] for which the determinant is [tex]0[/tex]. The determinant is [tex]0[/tex] if there is some linear dependence in the rows, by simple inspection/observation you can tell that by setting [tex]x=0,1,2,\ldots, n-1[/tex] you can make one of the rows the same as the first row, thereby creating a linear dependence, and forcing the determinant to be [tex]0[/tex]. So the [tex]n[/tex] solutions must be [tex]x=0,1,2,\ldots, n-1[/tex].

Hope this helped,

R. Baber,

[Guest]

Hey, it turns out to be a relatively easy problem. Thank you, R.Baber!