# Determinants

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• September 14th 2008, 07:58 PM
mivanova
Determinants
Hi, Thanks for the help before.
Can you help me with this new problem.
Define the following determinants of the matrices whose elements come from the ring F[x]:
δ1=1
δ2= | 0 -1|
|-1 x|
δ3= | 0 -1 0|
| 0 x -1|
|-1 0 x|
δ4= | 0 -1 0 0|
| 0 x -1 -1|
| 0 0 x -1|
|-1 0 0 x|
Continue in this fashion. δn is the determinant of an nxn matrix each of whose entries is either 0 or -1 or x according to the following rule. The diagonal entries are all equal to x except for the first diagonal entry which is 0. Each entry along the super diagonal, (i.e. just above the main diagonal) equals -1, as does the entry in the lower left-hand corner. All other entries are 0. Evaluate δn.
Thank you!
• September 15th 2008, 12:04 AM
NonCommAlg
Quote:

Originally Posted by mivanova
Hi, Thanks for the help before.
Can you help me with this new problem.
Define the following determinants of the matrices whose elements come from the ring F[x]:
δ1=1
δ2= | 0 -1|
|-1 x|
δ3= | 0 -1 0|
| 0 x -1|
|-1 0 x|
δ4= | 0 -1 0 0|
| 0 x -1 -1|
| 0 0 x -1|
|-1 0 0 x|
Continue in this fashion. δn is the determinant of an nxn matrix each of whose entries is either 0 or -1 or x according to the following rule. The diagonal entries are all equal to x except for the first diagonal entry which is 0. Each entry along the super diagonal, (i.e. just above the main diagonal) equals -1, as does the entry in the lower left-hand corner. All other entries are 0. Evaluate δn.
Thank you!

we have $\delta_1=1.$ suppose $n>1.$ so i guess this is how $\delta_n$ is defined: $\delta_n=\begin{vmatrix}0 & -1 & 0 & 0 & \cdots & 0 \\ 0 & x & -1 & 0 & \cdots & 0 \\ 0 & 0 & x & -1 & \cdots & 0 \\ . & . & . & . & \cdots & . \\
. & . & . & . & \cdots & . \\ . & . & . & . & \cdots & . \\ -1 & 0 & 0 & 0 & \cdots & x \end{vmatrix}.$
what you need to do is to multiply the first row by $x$ and add the result to the

second row to get rid of $x$ in the second row. then multiply the second row by $x$ and add the result to the third row to get rid of $x$ in the third row. continue until all $x$ are gone. thus:

$\delta_n=\begin{vmatrix}0 & -1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & -1 & 0 & \cdots & 0 \\ 0 & 0 & 0 & -1 & \cdots & 0 \\ . & . & . & . & \cdots & . \\
. & . & . & . & \cdots & . \\ . & . & . & . & \cdots & . \\ -1 & 0 & 0 & 0 & \cdots & 0 \end{vmatrix}$

$=(-1)^n \begin{vmatrix}-1 & 0 & 0 & \cdots & 0 \\ 0 & -1 & 0 & \cdots & 0 \\ 0 & 0 & -1 & \cdots & 0 \\ . & . & . & \cdots & . \\
. & . & . & \cdots & . \\ . & . & . & \cdots & . \\ 0 & 0 & 0 & \cdots & -1 \end{vmatrix}=(-1)^{2n-1}=-1$