Prove that the determinant of the matrix

is equal to .

Do I approach this with induction? where do I start?

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- October 22nd 2013, 02:46 PMvidomagruFinding the Determinant
Prove that the determinant of the matrix

is equal to .

Do I approach this with induction? where do I start? - October 22nd 2013, 03:38 PMemakarovRe: Finding the Determinant
Subtract the last row from every other row.

- October 22nd 2013, 04:57 PMHallsofIvyRe: Finding the Determinant
emakarov is suggesting a row operation. Do you know how row operations on a matrix change the determinant?

If you swap two rows, you multiply the determinant by -1.

If you multiply a row by a constant, you multiply the determinant by that constant.

If you add a multiple of a row to another row (such as adding 1 times the last two to each other row) does not change the determinant. - October 22nd 2013, 08:29 PMvidomagruRe: Finding the Determinant
THANK YOU! I would not have caught that trick. So here is what I did:

Which of course is a diagonal matrix so the determinant is the product of the diagonal terms, however the last one is 0, which throws this off. What did I do wrong?

I can see that the first terms give you but where does the last come from? - October 23rd 2013, 12:05 AMemakarovRe: Finding the Determinant
The hint was to subtract the last row from every row except the last one. Subtracting a row v from a different row w, (as HallsofIvy wrote, formally this is replacing w with the sum w + (-1)v) does not change the determinant. Subtracting a row from itself definitely changes the determinant in general: it turns it to zero!

- October 23rd 2013, 04:04 AMvidomagruRe: Finding the Determinant
- October 23rd 2013, 06:56 AMemakarovRe: Finding the Determinant
I'll borrow your wonderful LaTeX code.

Expanding the determinant along the last column (Laplace's formula), we get

.