Integration with matrices

Question

I have written two equations in matrix format as follows

$m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} \phi_1(t)\\ \phi_2(t)\\ \phi_3(t)\\ \phi_4(t)\\ \end{pmatrix} \tag1$

$n(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} \psi_1(t)\\ \psi_2(t)\\ \psi_3(t)\\ \psi_4(t)\\ \end{pmatrix} \tag1$

Is it possible to write an expression for the following questions with out multiplying the the constant matrices with variable matrices.? I imply a matrix form for integration. ( For example I can write $ \int m(t) dt ={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} \int \phi_1(t)dt\\ \int \phi_2(t)dt\\ \int \phi_3(t)dt\\ \int \phi_4(t)dt\\ \end{pmatrix} +C$ . I assume I am correct.)

Question

1. What could be the matrix format for $\int m(t)^2 \ dt $ in a similar matrix format as shown in the given example above?
2. What could be the matrix format for $\int m(t)n(t) \ dt $ in a similar format as shown in the given example above?