How Can I Solve a Complex Determinant Problem for an nxn Matrix?

[bmb2009]

Homework Statement

The problem is attached due to it not being able to copy over the given matrix..
Number 5.3

Homework Equations

The Attempt at a Solution

I tried to compute A+tI and then row reduce until A+tI was a triangle matrix and then multiply the diagnol entries but I got stuck.. any help would be great! thanks!

 

[Dick]

Homework Statement

The problem is attached due to it not being able to copy over the given matrix..
Number 5.3

Homework Equations

The Attempt at a Solution

I tried to compute A+tI and then row reduce until A+tI was a triangle matrix and then multiply the diagnol entries but I got stuck.. any help would be great! thanks!

You don't want to do it that way. They give you the hint to try row expansion (i.e. expansion by minors). I'd suggest you write down the matrices in the cases n=2 and n=3 and work out the determinants. Then you can probably make a guess for what the general case would look like. Now for the general case expand along the first row. There are only two nonvanishing entries. The minor for one is easy and the minor for the other looks a lot like a lower dimensional version of the same problem. Now use the other hint to use induction to prove your guess.

 

[bmb2009]

I've been trying that but we never covered co factor expansion in class so I'm trying to self teach this from internet resources...I see multiple examples for 3x3 matrices but I don't see any process to follow for nxn

 

[Dick]

I've been trying that but we never covered co factor expansion in class so I'm trying to self teach this from internet resources...I see multiple examples for 3x3 matrices but I don't see any process to follow for nxn

It works EXACTLY the same way. Go across the first row, multiply the element by the determinant of the matrix you get by striking out the row and column the element is in, multiply by a factor of +/-1 and sum. Did you start by working out the n=2 and n=3 cases? Do those first.

 

[bmb2009]

I apologize for the stupid questions.. but I don't know what you mean by the n=2, n=3 cases... my confusion is for a 3x3 matrix after you strike out the column and row the element is in you are left with 3 2x2 matrices... which the determinants are easy to compute for 2x2 but once you strike out the nxn and I have an (n-1)x(n-1) i don't know how to calculate the determinant of what's left..

 

[haruspex]

once you strike out the nxn and I have an (n-1)x(n-1) i don't know how to calculate the determinant of what's left..

There are only two nonzero terms in the first row, yes? So expanding by that row gives you two terms, each involving an (n-1)x(n-1) determinant. The first should look very like what you started with; the second will be zeroes except on the diagonal and the superdiagonal. Row or column manipulation will quickly evaluate the second. The first will lead to a kind of recurrence relation, so you could try induction. Trying 2x2 and 3x3 examples might help you understand how to do the induction.

 

[Dick]

I apologize for the stupid questions.. but I don't know what you mean by the n=2, n=3 cases... my confusion is for a 3x3 matrix after you strike out the column and row the element is in you are left with 3 2x2 matrices... which the determinants are easy to compute for 2x2 but once you strike out the nxn and I have an (n-1)x(n-1) i don't know how to calculate the determinant of what's left..

You are given a description of an nxn matrix. Use that pattern to figure out out what what it looks like in specific cases. If n=2 then the matrix (after adding tI) is [[t,a0],[-1,a1+t]], right? What's the determinant of that? You do 3x3. It's good practice. Try row expanding it along the first row and pay attention to what the cofactors look like. May help with the general case.