### Related Formula According to Cramer's Rule, for a system of linear equations to have infinitely many solutions, the main determinant Delta$\Delta$ and all directional determinants Delta_1, Delta_2, Delta_3$\Delta_1, \Delta_2, \Delta_3$ must equal zero simultaneously. ### Core Logic Set up the directional determinant equation Delta_3 = 0$\Delta_3 = 0$ by replacing the third column with the constant vector: Delta_3 = left| beginmatrix 2 & -1 & 4 \\ 5 & lambda & 12 \\ 100 & -47 & 212 endmatrix right| = 0$\Delta_3 = \left| \begin{matrix} 2 & -1 & 4 \\ 5 & \lambda & 12 \\ 100 & -47 & 212 \end{matrix} \right| = 0$ Expand the determinant along the first row: 2[212lambda - 12(-47)] - (-1)[5(212) - 12(100)] + 4[5(-47) - 100lambda] = 0$2[212\lambda - 12(-47)] - (-1)[5(212) - 12(100)] + 4[5(-47) - 100\lambda] = 0$ 2[212lambda + 564] + 1[1060 - 1200] + 4[-235 - 100lambda] = 0$2[212\lambda + 564] + 1[1060 - 1200] + 4[-235 - 100\lambda] = 0$ 424lambda + 1128 - 140 - 940 - 400lambda = 0$424\lambda + 1128 - 140 - 940 - 400\lambda = 0$ 24lambda + 48 = 0 implies lambda = -2$24\lambda + 48 = 0 \implies \lambda = -2$ ### Step 1: Solve for Mu using the main determinant Set the primary coefficient matrix determinant Delta = 0$\Delta = 0$ and substitute lambda = -2$\lambda = -2$: Delta = left| beginmatrix 2 & -1 & 1 \\ 5 & -2 & 3 \\ 100 & -47 & mu endmatrix right| = 0$\Delta = \left| \begin{matrix} 2 & -1 & 1 \\ 5 & -2 & 3 \\ 100 & -47 & \mu \end{matrix} \right| = 0$ Expand the determinant along the first row: 2[-2mu - 3(-47)] - (-1)[5mu - 3(100)] + 1[5(-47) - (-2)(100)] = 0$2[-2\mu - 3(-47)] - (-1)[5\mu - 3(100)] + 1[5(-47) - (-2)(100)] = 0$ 2[-2mu + 141] + [5mu - 300] + [-235 + 200] = 0$2[-2\mu + 141] + [5\mu - 300] + [-235 + 200] = 0$ -4mu + 282 + 5mu - 300 - 35 = 0$-4\mu + 282 + 5\mu - 300 - 35 = 0$ mu - 53 = 0 implies mu = 53$\mu - 53 = 0 \implies \mu = 53$ ### Step 2: Calculate the Target Value Substitute the values of mu$\mu$ and lambda$\lambda$ into the expression: mu - 2lambda = 53 - 2(-2) = 53 + 4 = 57$\mu - 2\lambda = 53 - 2(-2) = 53 + 4 = 57$ ### Pattern Recognition When solving systems of equations for infinite solution parameters, choosing a directional determinant that excludes one of the variables simplifies the problem into two separate single-variable equations. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants