Let $A$ and $B$ be two $n \times n$ real matrices. (In my application, $A$ and $B$ are $6\times 6$ traceless singular real matrices) I am interested in finding the smallest $T$ such that the integral $\int_0^T \exp(tA) \exp(tB)dt$ becomes singular.

If in the integrand there is only one exponential (such is the case when $A$ and $B$ commute) then there is a result by Kalmann-Ho-Narendra which states that the matrix $\int_0^T \exp(tA)dt$ is invertible if $T(\mu−\lambda) \ne 2k\pi i$ for any nonzero integer $k$, where $\mu$ and $\lambda$ are any pair of eigenvalues of $A$.

Let $A$ and $B$ be two $n \times n$ real matrices. (In my application, $A$ and $B$ are $6\times 6$ traceless singular real matrices) I am interested in finding the smallest $T$ such that the integral $\int_0^T \exp(tA) \exp(tB)dt$ becomes a singular matrix.

If in the integrand there is only one exponential (such is the case when $A$ and $B$ commute) then there is a result by Kalmann-Ho-Narendra which states that the matrix $\int_0^T \exp(tA)dt$ is invertible if $T(\mu−\lambda) \ne 2k\pi i$ for any nonzero integer $k$, where $\mu$ and $\lambda$ are any pair of eigenvalues of $A$.