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Suppose

$A(t) = \left\lbrack \begin{array}{rr} t & t^{4} \\ 0 & 6t \end{array} \right\rbrack.$
(a) Calculate $A^{-1}(t)$ and $\displaystyle\frac{d}{dt}\left( A^{-1}(t) \right)$ .

$A^{-1}(t) =$

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

$\displaystyle \frac{d}{dt}\left( A^{-1}(t) \right) =$

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

(b) Calculate $A'(t)$ and $\displaystyle -A^{-1}(t) A'(t) A^{-1}(t)$ .

$A'(t) =$

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

$\displaystyle -A^{-1}(t) A'(t) A^{-1}(t) =$

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

How are $\displaystyle \frac{d}{dt} \left( A^{-1}(t) \right)$ and $-A^{-1}(t) A'(t) A^{-1}(t)$ related?

You can earn partial credit on this problem.