Solving Square Matrix Similarity to Diagonal Matrix

[I like Serena]

Well, I believe we have the answer for (a), don't you?

As for (b), I believe we can use the fact that any diagonal matrix with only 3's and 5's on the diagonal should be a solution for the equation...

 

[micromass]

Well, I believe we have the answer for (a), don't you?

As for (b), I believe we can use the fact that any diagonal matrix with only 3's and 5's on the diagonal should be a solution for the equation...

Aah, thus (b) turns out to the following problem: for each k>8, prove that k can be written as the sum of 3's and 5's.

This should be fun

 

[I like Serena]

Aah, thus (b) turns out to the following problem: for each k>8, prove that k can be written as the sum of 3's and 5's.

This should be fun

Hey! What about k=8?

 

[micromass]

Hey! What about k=8?

Aaaaah, of course, k=8 is the most important one!

Hint for the OP: first try to write k=8,9,...,15 as the sum of 3's and 5's. The rest of the numbers are easy (so I claim)

 

[Maybe_Memorie]

I'm confused as to what k actually is

 

[I like Serena]

I'm confused as to what k actually is

Your problem states:

"(b) Show that for every positive integer k >= 8 there exists a matrix A
satisfying the above condition with tr(A) = k."


So let's start with k=8.
Can you find an nxn matrix A with tr(A)=8 that satisfies: A^2 - 8A + 15I = 0?

If you can, then the next question is:
Suppose k=9.
Can you find an nxn matrix A with tr(A)=9 that satisfies: A^2 - 8A + 15I = 0?

...

 

[Maybe_Memorie]

8 = 3 + 5
9 = 3 + 3 + 3
10 = 5 + 5
11 = 3 + 3 + 5
12 = 3 + 3 + 3
13 = 5 + 5 + 3
14 = 5 + 3 + 3 + 3
15 = 5 + 5 + 5

Every k>15 will be a sum of previous numbers >8, so therefore it is clear to see that there is a matrix with the required condition with tr(A)=k and k>8

 

[micromass]

12 = 3 + 3 + 3

Forgot a +3 here. But the rest is totally correct!