How to align the first line of the equation to the left?

Why align everything left ? I would recommend using multlinehere

documentclass{article}
usepackage{mathtools}
begin{document}
begin{multline*}
detbigg(lambda I -ABC(G_1circ G_2)bigg)
=
left[
begin{array}{ccc}
A(G_2)otimes Q(G_1) otimes I_{n_2} &  G^T 
 G & A(G_1)otimes Q(G_2)otimes I_{n_1}
end{array}
right].
end{multline*}
end{document}

Two possibilities:

(red lines indicate text area borders)

documentclass{article}
usepackage{mathtools}
%---------------- show page layout. don't use in a real document!
usepackage{showframe}
renewcommandShowFrameLinethickness{0.15pt}
renewcommand*ShowFrameColor{color{red}}
%---------------------------------------------------------------%

begin{document}
    begin{align*}
detbiggl(lambda I -ABC(G_1circ G_2)biggr)    
    & = begin{bmatrix}
A(G_2)otimes Q(G_1) otimes I_{n_2} 
    &  G^T  
G   & A(G_1)otimes Q(G_2)otimes I_{n_1}
        end{bmatrix}.
    end{align*}
or with use of the verb+MoveEqLeft+ defined in the verb+mathtools+ package:
    begin{align*}
MoveEqLeft[4]detBigl(lambda I -ABC(G_1circ G_2)Bigr)    
    & = begin{bmatrix}
A(G_2)otimes Q(G_1) otimes I_{n_2}
    &  G^T  
G   & A(G_1)otimes Q(G_2)otimes I_{n_1}
        end{bmatrix}.
    end{align*}

You only have to declare in the code at least two alignment columns. The flalign environment doesn't align on the left automagically.

I took the liberty to simplify your code: as I didn't see why an array with 20 columns – I replaced it with the simpler bmatrix environment. I also didn't see why the huge parentheses for the argument of det, so I replaced the pair bigg( ... bigg) with bigl( ... bigr) (note the final l and r).

documentclass{article}
usepackage{amsmath}
usepackage{showframe}
renewcommand{ShowFrameLinethickness}{0.2pt}

begin{document}

begin{flalign*}
detbigl(lambda I & -ABC(G_1circ G_2)bigr) & &
[0.5ex]
 & =
begin{bmatrix}
A(G_2)otimes Q(G_1) otimes I_{n_2} && G^T
 [1ex]
 G &&
A(G_1)otimes Q(G_2)otimes I_{n_1}
end{bmatrix}
end{flalign*}

end{document}