Hi everyone.

I've been doing determinants at school, however I've come across one which is making my head spin.

We've been told to simplify them as much as possible, adding or subtracting rows, columns together, taking factors out, etc to get as many 1's and 0's as possible so we can take out factors to construct a determinant.

However, I am really stuck on this bad boy.

I've tried adding the bottom row to the middle, then to the top, but I get an extra a^2, subtracting columns, rows etc, but everything I've tried so far has left me with just one factor out of place!

I would appreciate some help and guidance with this, please. I realise that it's going to look ugly before any nice happens, but I can only get to dead ends.

Many thanks for your time,

Sparky

 

Have you learned to use the method of expansion by diagonals?

http://www.jcoffman.com/Algebra2/ch4_6.htm

 

No, although I can follow the instructions there.

If anyone can spot how to find the determinant by adding/subtracting rows and columns, I'd be very impressed.

Sparky.

P.S. Thanks for the link!

 

No, although I can follow the instructions there.

If anyone can spot how to find the determinant by adding/subtracting rows and columns, I'd be very impressed.

Sparky.

P.S. Thanks for the link!

Once you get into 3rd and 4th power determinants, I don't think it's as simple as adding/subtracting rows and columns. I'm not sure if this is the method you're looking for, but it works. I believe it's called the laplace expansion method, but I could be thinking of something else:

| a  b  c |     | e f |     | d f |     | d e |
| d  e  f | = a | h i | - b | g i | + c | g h |
| g  h  i |

Can you see how that works? You basically have to split the array up into smaller parts, and multiply it by a corresponding value in the first row. You can add them together (switching the signs each time, though--that's very important) and solve it that way. If you look at the example, you'll see the pattern.

Here's an example I just found of one being done using this method: