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Suppose

$A''(t) = \left\lbrack \begin{array}{rr} 1 & t \\ 0 & 0 \end{array} \right\rbrack, \ \ \ A(0) = \left\lbrack \begin{array}{rr} 5 & -3 \\ 2 & -1 \end{array} \right\rbrack, \ \ \ A(1) = \left\lbrack \begin{array}{rr} -3 & 1 \\ -3 & 3 \end{array} \right\rbrack.$
Determine the general form of $A(t)$ by constructing antiderivatives as needed and imposing any given constraints.

$A(t) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$			$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$