Finding a bound on the norm of a matrix

Question

first of all I'm sorry but I don't know how to write expressions, so I used images to explain my problem.

I have to find the following parameters: alpha and beta in order that the following bound holds:

where the matrix A is the following one

According to the definition of norm of a matrix, I should find the max eigenvalue of this matrix A.

First of all, I have computed:

Then looking at the solution, I have seen that, the lower 2 × 2 block on the diagonal of this A^TA matrix has been called B and the maximum eigenvalue of AT A is:

Question 1: Is the maximum eigenvalue of a matrix always computed in this way? (Cosidering the trace). I usually compute the det(lambda*I-ATA) and then I compute the max eigenvalue

Then in the solution, there was written: Since we are looking for a bound on the norm of A(q), we can write the chain of inequalities

Question 2: Is the bound on the norm of a matrix, always less than the trace of the same matrix?

Question 3: I didn't understand from where the highlighted expressions come up. Could anyone help me?

The final result is the following (but I don't care, I would like to understand the steps):