Prove that the determinant of the matrix
is equal to .
Do I approach this with induction? where do I start?
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.
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?
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!
I'll borrow your wonderful LaTeX code.
Expanding the determinant along the last column (Laplace's formula), we get
.