Question

If \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \] then the value of \[ \left( \frac{24}{x} + \frac{24}{y} \right) \] is:

• A. 7
• B. 6
• C. 8
• D. 18

Hint:

When solving matrix equations, compare corresponding elements to derive equations for unknowns. Simplify expressions by substituting these values back into the problem, ensuring all calculations align with the matrix structure.

Answer 1

The Correct Option is D

Given the matrix equality: \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \]

Step 1: Determine \( x \) and \( y \)
From the equality, we have the system of equations: \[ x + y = 6 \quad \text{and} \quad xy = 8. \] This implies \( x \) and \( y \) are the roots of the quadratic equation: \[ t^2 - (x + y)t + xy = 0. \] Substituting the values: \[ t^2 - 6t + 8 = 0. \] Factoring the quadratic: \[ (t - 2)(t - 4) = 0. \] Therefore, the possible values for \( x \) and \( y \) are 2 and 4 (in any order). 

Step 2: Calculate \( \frac{24}{x} + \frac{24}{y} \) 
Assuming \( x = 2 \) and \( y = 4 \): \[ \frac{24}{x} + \frac{24}{y} = \frac{24}{2} + \frac{24}{4}. \] Simplifying the expression: \[ \frac{24}{2} + \frac{24}{4} = 12 + 6 = 18. \]