How can I make this matrix a triangular one?

[Hernaner28]

Hi. How can I reduce this matrix into a triangular one so I can calculate the determinant easily.

\displaystyle\left( {\begin{array}{*{20}{c}}<br /> b&amp;1&amp;1&amp;1&amp;1 \\ <br /> 1&amp;b&amp;1&amp;1&amp;1 \\ <br /> 1&amp;1&amp;b&amp;1&amp;1 \\ <br /> 1&amp;1&amp;1&amp;b&amp;1 \\ <br /> 1&amp;1&amp;1&amp;1&amp;b <br /> \end{array}} \right)

I've tried but I cannot make a triangular form...

Thanks!

 

[DryRun]

Reduce to row echelon form by Gaussian elimination.

 

[Hurkyl]

Have you tried computing the determinant by other means?

 

[DryRun]

If you need to find the determinant, try the cofactor expansion by the first row.

 

[Hernaner28]

I could! I subtracted row 1 from all rows and then I subtracted the all columns from the first.

 

[Hurkyl]

The cofactor expansion was the one I was thinking of earlier.

Yet another approach is to write the matrix as the sum A + (b-1) I, where A is the matrix of all 1's, and I is the identity matrix. A is diagonalizable, and you can find its eigenvalues without too much trouble -- and so you should also be able to find the eigenvalues of the sum!