Matrix diagonalisation

For the first part you could have preceded with
Qe = le
where l is an evalue and e and evector, then put it in matrix form. Similarly you could mention the |E|!=0 part and which part that impacts on. But it is a couple of lines and pretty much standard textbook derivation.

Is it true that |E| =\ 0 for all ( not necessarily symmetric) matrices? I know for symmetric matrices they can be made all orthogonal but what about non symmetric matrices?

Theres an example towards the bottom for a nonsymmetric 2*2 matrix
https://www.statlect.com/matrix-algebra/linear-independence-of-eigenvectors
so really youre assuming the evectors are linearly independent or |E|!=0

Ok, so is it important to mention that the equation only holds for linearly independent eigen vectors?

Id have mentioned it when you assumed E^(-1) existed / that |E|!=0. You wouldnt probably have to relate it to linearly independent evectors, just that the matrix is invertible because the det is non-zero. However, theres no harm in a bit more understanding.

I See, for part ii do you think this is enough working:

Seeing as it is a 5 mark question, this working honestly looks worth 2 marks, is there more to it?

The top like is meant to have k on the outside of the brackets not as an exponent for D