In the lab an analytical Chemist measures the fractional Pb, Fe, Cu and Mn content on a weight for weight basis for three ores, results are shown in below Figure.

If the Chemical Ore Analysis are mixed together so there are x_{1} kg of Ore 1, x_{2} kg of Ore 2, and x_{3} kg of Ore 3, the amount of each metal in the mixture can be computed as shown in below Figure.

Firstly, rewrite the above series of equations in matrix form A = FX where F is a 4×3 matrix containing the fractional elemental compositions; A and X are column matrices containing the total amount of each element and the quantities of each Chemical Ore Analysis respectively. Secondly, calculate the percent weight for weight (%w/w) of each element in a 10 kg mixture of the ores containing 4 kg of Ore 1, 3 kg of Ore 2 and 3 kg of Ore 3.

**Solution**

Firstly, the Analytical Chemists results written in matrix form are given below, where A = FX.

For the 10 kg of the mixed Chemical Ore Analysis, the product of a 4×3 matrix and a 3×1 matrix is a 4×1 matrix,

The 10 kg of mixture has Pb = 19%, Fe = 26%, Cu = 36% and Mn = 19%.