Consider the matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ indicated by the output grids below, as well as the shaded green areas in each grid: [asy] size(500); //Draws grid lines with the given parameters: void gridLines(int xmin, int xmax, int ymin, int ymax) { for(int i = ymin+1; i< ymax; ++i) { draw((xmin,i)--(xmax,i), mediumgray); } for(int i = xmin+1; i< xmax; ++i) { draw((i,ymax)--(i,ymin), mediumgray); } } //Gives the maximum line that fits in the box. path maxLine(pair A, pair B, real xmin, real xmax, real ymin, real ymax) { real s = abs(xmax) + abs(xmin) + abs(ymax) + abs(ymin) + abs(A.x - xmin) + abs(A.x-xmax) + abs(A.y - ymin) + abs(A.y - ymax); pair[] endpoints = intersectionpoints(A+2sunit(B-A) -- A-2sunit(B-A), (xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle); if (endpoints.length >= 2) return endpoints[0]--endpoints[1]; else return nullpath; } pair O, O2, O3, O4, AI, AJ, BI, BJ, CI, CJ, DI, DJ; pair tailAI, tailAJ, tailBI, tailBJ, tailCI, tailCJ, tailDI, tailDJ; path shapeA, shapeB, shapeC, shapeD; AI = dir(50); AJ = 2dir(200); BI = 2dir(20); BJ =dir(-70); CI = sqrt(3)dir(20); CJ =sqrt(6)dir(65); DI = 2dir(-30); DJ = 4dir(-60); //Change this to scale the grid: int big = 5; O = (0,0); O2 = (2big + 2, 0); O3 = (4big + 4, 0); O4 = (6big + 6, 0); //Matrix A output grid: shapeA = -4AI - 3AJ -- -4AI + AJ -- -AI+AJ -- -AI - 3AJ --cycle; fill(shapeA, green); for (int i = -15; i < 15; ++i) { draw(maxLine(iAI, iAI + AJ, -big, big, -big,big), linewidth(0.4)); draw(maxLine(iAJ, iAJ + AI, -big, big, -big, big), linewidth(0.4)); } draw(tailAI --tailAI + AI, red+linewidth(0.8), Arrow(7)); draw(tailAJ-- tailAJ + AJ, blue+linewidth(0.8), Arrow(7)); label("$\mathbf{A} \mathbf{i}$", AI, N, UnFill); label("$\mathbf{A} \mathbf{j}$", AJ, NW, UnFill); //Matrix B output grid: shapeB = -2BI - 3BJ -- -2BI - BJ -- BI- BJ -- BI - 3BJ --cycle; fill(shift(O2)shapeB, green); for (int i = -10; i <10; ++i) { draw(shift(O2)maxLine(iBI, iBI + BJ, -big, big, -big,big), linewidth(0.4)); draw(shift(O2)maxLine(iBJ, iBJ + BI, -big, big, -big, big), linewidth(0.4)); } draw(shift(O2)(tailBI --tailBI + BI), red+linewidth(0.8), Arrow(7)); draw(shift(O2)(tailBJ-- tailBJ + BJ), blue+linewidth(0.8), Arrow(7)); label("$\mathbf{B} \mathbf{i}$", shift(O2)BI, E, UnFill); label("$\mathbf{B} \mathbf{j}$", shift(O2)BJ, S, UnFill); //Matrix C output grid: shapeC = shift(-CI)(-CI -- O-- 2CJ -- 2CJ + CI -- -CJ + CI -- -CJ - CI -- cycle); fill(shift(O3)shapeC, green); for (int i = -10; i <10; ++i) { draw(shift(O3)maxLine(iCI, iCI + CJ, -big, big, -big,big), linewidth(0.4)); draw(shift(O3)maxLine(iCJ, iCJ + CI, -big, big, -big, big), linewidth(0.4)); } draw(shift(O3)(tailCI --tailCI + CI), red+linewidth(0.8), Arrow(7) ); draw(shift(O3)(tailCJ-- tailCJ + CJ), blue+linewidth(0.8), Arrow(7) ); label("$\mathbf{C} \mathbf{i}$", shift(O3)CI, 2E, UnFill); label("$\mathbf{C} \mathbf{j}$", shift(O3)CJ, 2E, UnFill); //Matrix D output grid: shapeD = +DI -- O-- DJ -- DJ - DI -- -DJ - DI -- -DJ + DI -- cycle; fill(shift(O4)shapeD, green); for (int i = -10; i <10; ++i) { draw(shift(O4)maxLine(iDI, iDI + DJ, -big, big, -big,big), linewidth(0.4)); draw(shift(O4)maxLine(iDJ, iDJ + DI, -big, big, -big, big), linewidth(0.4)); } draw(shift(O4)(tailDI --tailDI + DI), red+linewidth(0.8), Arrow(7) ); draw(shift(O4)*(tailDJ-- tailDJ + DJ), blue+linewidth(0.8), Arrow(7) ); label("$\mathbf{D} \mathbf{i}$", shift(O4)*DI, E, UnFill); label("$\mathbf{D} \mathbf{j}$", shift(O4)*DJ, SE, UnFill); [/asy] If each shaded green area is equal to $2$, what are [| \det (\mathbf{A})|, |\det (\mathbf{B})|, |\det (\mathbf{C})|, |\det (\mathbf{D})|]in that order?