Using Cramer’s Rule to solve a system of linear equation

I have trouble solving the following question: Question

How I approached it initially was as follow:

1) Since the questions asks for x3 only, I replaced the x3 column with the particular solution. Step 1

2) Cramer’s Rule asks of us to divide the det(Substituted Matrix in Step 1) by the det(Initial Matrix) Step 2

3) To find the det of each 4×4 matrix I did C(1,1)(det(3×3 sub Matrix)). So 1(det(3×3 sub Matrix)). I evaluated the 3×3 sub Matrix as follow: 3×3 sub Matrix The red arrows are added and then subtracted to the addition of the blue arrows. Thus det(Substituted Matrix) = 3aei + 3bfg + 2cdh – 3ceg – 2afh – 3bdi.

4) I did step 3 for the initial Matrix as well and got det(Initial Matrix) = aei + bfg + cdh – ceg – afh – bdi

Now, I am unsure how to work around the determinants I just got. I should factor out something and then cancel the rest, but I don’t see what. Did I miss something?

Including the mean in differential equation

I have a differential equation in the form: $ $ \frac{1}{C} \frac{dC}{dt} = a – C – \mu_C $ $

where C is a random variable. Is it possible to derive an exact solution to this equation to get the time-dependence of the variable C and its distribution? I know that we can do so without the average term at the end but am not sure how to deal with the mean due to its time dependence.

Solving equation with multi variable matrix input

Background, here are the equations that I am trying to solve: Full Equations

Where R, E1, E2, V1, V2, P are all user inputs. X/A goes from -2 to 2 and Z/A goes from 0 to -2. Below is the code that I have so far. I created a list of inputs. Then created two arrays for the x and z inputs. The last is where I am having trouble. I’m trying to create a code such that it will hold a value for X constant in SX, SZ, and TXZ and plug in all the values for Z. Then move to the next value for X and plug all the values in for all the Z. The end goal is to create a density plot that for SX, SZ, and TXZ. Thank you!

R = .1; E1 = 200*10^9; E2 = 550*10^9; P=1000; V1 = 0.3; V2 = 0.3; E = 1/(((1-(V1^2))/E1)+((1-(V2^2))/E2)); A = ((.75*P*R)/(1.61172*10^11))^(1/3); X = Range[-2 A, 2 A, 0.01*3*A]; Z = Range[0,-2 A, 0.005*3*A];  ZZ = ConstantArray[Z[[Range[Length[Z]]]], Length[X]]; XX = ConstantArray[X[[Range[Length[X]]]],Length[Z]];  For[i=1,i=Length[XX], For[j=1,j = Length[ZZ], M = Sqrt(.5*(((A^2-i^2+j^2)^2+4*i^2*j^2)^(.5)+(A^2-i^2+j^2))) N = Sqrt(.5*(((A^2-i^2+j^2)^2+4*i^2*j^2)^(.5)-(A^2-i^2+j^2))) SX = (-P/A)*M*((1-((j^2+N^2)/(M^2+N^2)))-2*N) SZ = (-P/A)*M*((1-((j^2+N^2)/(M^2+N^2)))) SY = V1*(SX+SZ) TXZ = (-P/A)*N*((M^2-j^2)/(M^2+N^2)), DensityPlot[SX/P,XX/A,ZZ/A]  ] ]