How do you calculate determinants and eigenvalues in Mathematica?

[Lambda96]

TL;DR

How to calculate eigenvalues, eigenvectors, determinants and inverses of a general nxn matrix

Hi,

In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment.

Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with Mathematica.

I was able to calculate the task a to c. Here is the solution for task b and the Matrix for the case $n=5$.

I have now problems with the task d, where I should calculate the determinants, eigenvalues, etc for the general case n or rather give a formula.

For the determinant as an example I proceeded as follows and unfortunately I get the following error:

Unfortunately I don't know how to fix the error or is my initial equation already wrong?

 

[DrClaude]

You can't leave n undefined. Mathematica can't find the rule for arbitrary n by itself. You have to calculate the result for explicit values of n, and then find by yourself the general rule.

 

[renormalize]

As DrClaude says, $n$ must be specified explicitly:

The pattern is pretty obvious.

 

[Lambda96]

Thanks DrClaude and renormalize for your help

Thanks also renormalize for the trick with plotting n objects in Mathematica, so I could directly plot the eigenvalues from $n=1$ to $n=10$ without repeating the calculation 10 times

I could now recognize corresponding patterns for the other values