/sci/ - Science & Math

can anyone how this inverse came about? Whenever I search about "inverse of a matrix of functions/solutions" I dont get any useful results

I get that the matrix of coefficients is [ 6 -3 2; -1 1 1; -5 1 1] and the inverse of THAT matrix is [ 0 1/4 -1/4; -1/5 4/5 -2/5; 1/5 9/20 3/20] and I suppose the LITERAL inverse (reciprocal) of [e^-t e^-2t e^3t] would be [e^t e^2t e^-3t] (I'm assuming that, please correct me if Im wrong. Idk how to take the inverse of a vector or a matrix of functions).
And since X = matrix of coefficients * [e^-t e^-2t e^3t] I assumed that its inverse would be the inverse of the coefficients * [e^t e^2t e^-3t] but apparently thats not the case according to pic related since matrix multiplication doesnt work that way
please help