Is a matrix with characteristic polynomial $t^2 +1$ invertible?

The eigenvalues of the matrix are all roots of the characteristic polynomial.

A square matrix is invertible if and only if $0$ is not an eigenvalue of the matrix.

Therefore, a square matrix is invertible if and only the constant term of its characteristic polynomial is <fill in the blank>

Yes it is. In fact $A^{-1}=-A$.

$A$ is a square matrix and $\det A=1$ so it is invertible.