Step 1: Define the characteristic equation. The characteristic equation is derived from the determinant of \( A - \lambda I \), leading to \( \lambda^2 - 8\lambda + 7 = 0 \).
Step 2: Calculate the determinant and simplify. The determinant simplifies to \( \lambda^2 - 8\lambda + 7 \), which factors to find the eigenvalues.
Step 3: Solve for \( \lambda \). Using the quadratic formula, we find the solutions to be \( \lambda = 7 \) and \( \lambda = 1 \), confirming the eigenvalues of the matrix.
Let $ A = \begin{bmatrix} 2 & 2 + p & 2 + p + q \\4 & 6 + 2p & 8 + 3p + 2q \\6 & 12 + 3p & 20 + 6p + 3q \end{bmatrix} $ If $ \text{det}(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n, \, m, n \in \mathbb{N}, $ then $ m + n $ is equal to:
Consider the balanced transportation problem with three sources \( S_1, S_2, S_3 \), and four destinations \( D_1, D_2, D_3, D_4 \), for minimizing the total transportation cost whose cost matrix is as follows:
where \( \alpha, \lambda>0 \). If the associated cost to the starting basic feasible solution obtained by using the North-West corner rule is 290, then which of the following is/are correct?
Then, which one of the following is TRUE?
Length of the streets, in km, are shown on the network. The minimum distance travelled by the sweeping machine for completing the job of sweeping all the streets is ________ km. (rounded off to nearest integer)