From the equality of the matrices: \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \]
Step 1: Use the relationship between \( x \) and \( y \)
The given equations are: \[ x + y = 6 \quad \text{and} \quad xy = 8. \] These are the sum and product of the roots of a quadratic equation. Let the quadratic equation be: \[ t^2 - (x + y)t + xy = 0. \]
Substitute \( x + y = 6 \) and \( xy = 8 \): \[ t^2 - 6t + 8 = 0. \]
Factorize the equation: \[ t^2 - 6t + 8 = (t - 2)(t - 4) = 0. \] Thus, \( x = 2 \) and \( y = 4 \) (or vice versa).
Step 2: Compute \( \frac{24}{x} + \frac{24}{y} \)
Substitute \( x = 2 \) and \( y = 4 \): \[ \frac{24}{x} + \frac{24}{y} = \frac{24}{2} + \frac{24}{4}. \] Simplify: \[ \frac{24}{2} + \frac{24}{4} = 12 + 6 = 18. \]
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?
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is: