# How can I make this matrix a triangular one?

1. Apr 29, 2012

### Hernaner28

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

$$\displaystyle\left( {\begin{array}{*{20}{c}} b&1&1&1&1 \\ 1&b&1&1&1 \\ 1&1&b&1&1 \\ 1&1&1&b&1 \\ 1&1&1&1&b \end{array}} \right)$$

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

Thanks!

2. Apr 29, 2012

### sharks

Reduce to row echelon form by Gaussian elimination.

3. Apr 29, 2012

### Hurkyl

Staff Emeritus
Have you tried computing the determinant by other means?

4. Apr 29, 2012

### sharks

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

5. Apr 29, 2012

### Hernaner28

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

6. Apr 29, 2012

### Hurkyl

Staff Emeritus
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!