Linear Algebra - Determinant Proof

[erok81]

Homework Statement

Consider the n x n determinant in which each entry on the main diagonal is a 2, each entry on the two adjacent diagonals is a 1, and every other entry is a zero.

Expand along the first row to show that B_n=2B_n-1 - B_n-2

Homework Equations

n/a

The Attempt at a Solution

I drew out a matrix, B_n...see my crossed out example. I am not sure how to type out a matrix using latex.

Also in my image you can see where I started the cofactor expansion along row 1. After a few entries, I realized this isn't even close to correct, so I stopped.

Now I am at a loss of how to proceed. We got some guidance from the professor, but I don't quite understand still. I've included his comments below.

Write out determinant B_n on paper using "..." notation. At least six (6) columns
should be written, to give enough detail. Then use cofactor expansion along row 1
to produce 2 determinants of order n-1. Finally, use column 1 cofactor expansion
on the second determinant to produce a determinant of order n-2. All of this can
fit one one sheet of paper.

Any direction would be appreciated.

 

[Inferior89]

It seems you haven't used this information at all:
"each entry on the main diagonal is a 2, each entry on the two adjacent diagonals is a 1, and every other entry is a zero."

 

[erok81]

Oh *****.

That's where the n-1 values come from. Ok...let's try this again.

That was a pretty stupid oversight.

 

[erok81]

I guess where I am lost is he says to do it in dimension n and use the notation "...", examples won't count. If I use the original matrix I have all numbers, no n values, and no "..." notation.

Not to mention in the instructions it says it takes one page.

For me to write out b_n, b_n-1, b_n-2, and cofactor expansion for row one on each of the matrices, I am at three pages now. So there is no way what I am doing is right.

I also threw all three matrices into maple, solved for their determinants, tried them in the "proof" formula that is given and didn't get the right answer there either...-19=7.

 

[Inferior89]

The n value determines the size of the matrix. Use cofactor expansion along a suitable coloumn or row to find the determinant in terms of B_(n-1) and B_(n-2).

Your matrix should look something like this

$$\begin{array}{cccccc} 2 & 1 & 0 & 0 & 0 & \ddots \\ 1 & 2 & 1 & 0 & 0 & \ddots \\ 0 & 1 & 2 & 1 & 0 & \ddots \\ \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \end{array}$$

with n rows and columns.

 

[erok81]

Okay, that makes more sense.

We didn't really cover this in class, or what n-1 means, but now that you say that, it makes perfect sense.

Round three...

 

[erok81]

You are a genius!

I haven't written it out but I threw the whole thing, correct this time into maple. Now I get the correct answers.

Thank you.

Now I'll give it a go on paper.

 

[erok81]

It worked out. Took me four pages, but it got it.

Thanks again for the help.