With reference to the conventional Cartesian $$(x,y)$$ coordinate system, the vertices of a triangle have the following coordinates: $$\,\left( {{x_1},{y_1}} \right) = \left( {1,0} \right);\,\,\,\left( {{x_2},{y_2}} \right) = \left( {2,2} \right);\,\,\,$$ and $$\,\left( {{x_3},{y_3}} \right) = \left( {4,3} \right).$$ The area of the triangle is equal to

The sum of Eigen values of the matrix, $$\left[ M \right]$$
is where $$\left[ M \right] = \left[ {\matrix{ {215} & {650} & {795} \cr {655} & {150} & {835} \cr {485} & {355} & {550} \cr } } \right]$$

Given the matrices $$J = \left[ {\matrix{ 3 & 2 & 1 \cr 2 & 4 & 2 \cr 1 & 2 & 6 \cr } } \right]$$ and $$K = \left[ {\matrix{ 1 \cr 2 \cr { - 1} \cr } } \right],\,\,$$ the product $${K^T}JK$$ is ______.

For a sample of water with the ionic composition shown in the figure below, the carbonate and noncarbonate hardness concentrations (in mg/l as $${\mathrm{CaCO}}_3$$), respectively are: