Evaluation of determinant

Linear Algebra
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Tolaso J Kos
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Evaluation of determinant

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Post by Tolaso J Kos »

Let \(a_1, a_2, \; \dots, a_9 \) be real numbers. Evaluate the \( 10 \times 10 \) determinant:

$$\begin{vmatrix}
-1 & 1 &1 &1 &1 &1 &1 &1 &1 &a_1 \\
1& -1 & 1 & 1& 1 & 1& 1 & 1 &1 &a_2 \\
1&1 & -1 &1 &1 &1 & 1 &1 & 1 &a_3 \\
1&1 &1 & -1 &1 &1 &1 &1 &1 &a_4 \\
1 &1 &1 &1 &-1 &1 &1 &1 &1 &a_5 \\
1&1 &1 &1 &1 &-1 &1 &1 &1 &a_6 \\
1& 1 & 1 &1 &1 &1 &-1 &1 &1 &a_7 \\
1& 1 & 1 &1 &1 &1 &1 &-1 &1 & a_8\\
1&1 &1 &1 &1 &1 &1 &1 & -1& a_9\\
a_1&a_2 &a_3 &a_4 &a_5 &a_6 &a_7 &a_8 &a_9 & 0
\end{vmatrix}$$
Hidden Message
I am looking for a neat solution. Probably permutations?
Imagination is much more important than knowledge.
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