I had to find a matrix 2×2 and 4×4 that have no eigenvalues, for the 2×2 it was not that hard to do $a_{11}= 0$ $a_{12}= 1$ $a_{21}= -1$ $a_{22}= 0$ so that the possible eigenvalues are $det(xId-A)=x^2+1=x={-i,i}$, where A is the matrix with the $a_{ij}$ entries.
But I was not able to get to a 4×4 matrix
Best Answer
For a $3 \times 3$ real matrix, this does not exist.
Indeed, the characteristic polynomial of a $3 \times 3$ matrix is a polynomial of degree $3$, which has to vanish in $\mathbb{R}$. So there always exists an eigenvalue.
For a $4 \times 4$ matrix, consider the matrix
\begin{pmatrix} 0 & 0 & 0&-1\\ 1 & 0 & 0& 0\\ 0 & 1 & 0 & 0\\ 0& 0& 1& 0 \end{pmatrix}
Its characteritic polynomial is $X^4 +1$.