Suggest matrix $A$ with a maximal singular value of $5$

Question

Suggest matrix $A \in M_{4x4}$ such that :

1. $\mbox{rank}(A) = 2$

2. $A$ is not diagonal

3. maximal singular value of $A$ is $5$

I have tried to guess some symmetric matrices without any luck .

Answer 1

Let $D=diag(5,1,0,0)$, that way, the rank of $D$ would be $2$.

$QDQ^T$ would satisfy the first and the third condition if $Q$ is an orthogonal matrix.

In particular, you can pick $Q$ to be the hadamard matrix. I would leave the explicit construction to you.