Algebra in chemical engineering Could you provide an answer with explanations of the steps? I can post it multiple times

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Algebra in chemical engineering
Could you provide an answer with explanations of the steps? I can post it multiple times but the explanation is important. The excercise to solve is on picture number 3.
Spectral unmixing (Block 2) In absorption spectroscopy, a certain frequency of light is sent through a material that we wish to analyse. Depending on the composition of the material, certain frequencies of light will be absorbed more than others. Figure 1 shows the spectra of three materials. In spectral unmiring, we wish to determine the individual concentrations of the chemical components from the spectrum of a mixed solution of the materials, whose individual spectra are known. An example could be the determination of the concentrations of fructose, lactose, and ribose in a sugar solution. Individual spectra 140 120 - 100 80 Artimarts نامہ بدرالس 60 Licha hu hom 40 Inhamuth 0 1600 1500 1400 1300 1200 1100 700 500 500 400 300 200 1000 900 800 Ramanoffet om Figure 1: Spectral measurements from fructose, lactose, and ribose. Selected frequencies corresponding to candidates for pure component spectra of the three materials are also shown. The figure stems from 1, side 55). The absorption in a solution depends on the concentrations of the individual materials, on the frequency of the light, and on how far the light has to travel through the material In Figure 1, the dashed lines show frequencies where only one of the three materials absorb light - the pure component spectra. These frequencies are used during spectral unmixing We have performed n experiments on (sent light through) a solution of m materials in various solutions and p frequencies. The results of the experiments is logged in an nx p-matrix D where each row represents an experiment with absorption at the different frequencies. Our goal is to determine the concentration of the materials. "The Lambert-Beer law is a linear physical model stating that the absorption of light in a material is proportional to its concentration. The coefficient of proportionality depends on the material and the frequency of the light (combined in a so-called molar attenuation coefficient), as well as the distance of material through which the light travels.
The process can be described via two real matrices C and S. C is an n x m-matrix containing the unknown concentrations of the materials, such that each column corres- ponds to a material, and each row corresponds to an experiment with a different solution of materials. S is a known p x m-matrix with so-called pure component spectra consisting of coefficients of proportionality at given frequencies. These frequencies are chosen in such a way that only one of the materials absorbs light at any given frequency In total, the measurements are given by an n x p-matrix D: D= CST From this it is seen that DS = CSTS. If the matrix STS is invertible, we can use this to determine C from D and S since C = DS(STS) (1) in this case. We therefore determine the cases where STS is invertible. In practice, there will be measurement errors in D, which makes it more difficult to approximate C than what is done in (1), but we ignore these error in this workshop to make it easier to do the calculations by hand. It must be noted that within mathematics, spectrum is used to denote eg.. the set of eigenvalues of a matrix. This is not the same terminology as used in physics and chemistry in relation to spectroscopy.
Exercise 1 0 In this exercise, we consider a toy example with specific matrices D and S. To make hand calculation easier, the matrix values do not come from a spectroscopy analysis. Thus, let 2 1 - 4 2 2 0 2 3 D= and S= 1 - 1 2 6 1 2 2 -2 -2 4 0 -6 (i) Show by direct calculation that the matrix STS is given by 9 -1 -2 STS -1 9 2 2 2 8 (ii) Show that v1 = [1, -1, 2)" and v2 = [1, -1, -1)". are eigenvectors of STS. What are the corresponding eigenvalues? (iii) It turns out that the characteristic polynomial for STS is given by -(1-12)(x-8)(1-6). Use this to determine the remaining eigenvalues for STS, and determine the only (why can there be only one?) remaining eigenvector.