<p>Find non-zero values of <span>\(x\)</span> satisfying the matrix equation:<br>\[x\begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2\begin{bmatrix} 8 & 5x \\ 4 & 4x \end{bmatrix} = 2\begin{bmatrix} x^2+8 & 24 \\ 10 & 6x \end{bmatrix}\]</p>
Step-by-Step Solution
Key Concept: Expand the scalar multiplication on both sides, then equate corresponding matrix elements to get polynomial equations. The consistency of equations across different matrix positions determines the valid solution.
<p><strong>Step 1: Expand scalar multiplication on left side</strong></p><p>x·[2x, 2; 3, x] = [2x², 2x; 3x, x²]</p><p>2·[8, 5x; 4, 4x] = [16, 10x; 8, 8x]</p><p>Left side = [2x² + 16, 2x + 10x; 3x + 8, x² + 8x]</p><p><strong>Step 2: Expand right side</strong></p><p>2·[x² + 8, 24; 10, 6x] = [2x² + 16, 48; 20, 12x]</p><p><strong>Step 3: Equate corresponding elements</strong></p><p>Element (1,1): 2x² + 16 = 2x² + 16 ✓ (always true)</p><p>Element (1,2): 2x + 10x = 48 → 12x = 48 → <strong>x = 4</strong></p><p>Element (2,1): 3x + 8 = 20 → 3x = 12 → <strong>x = 4</strong></p><p>Element (2,2): x² + 8x = 12x → x² - 4x = 0 → x(x - 4) = 0 → <strong>x = 4</strong> (non-zero)</p><p><strong>Step 4: Verify consistency</strong></p><p>All three independent equations give x = 4, confirming this is the unique non-zero solution.</p><p>∴ <strong>Answer: x = 4</strong></p>
Correct Answer: x = 4