If then x is equal to

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

maximum and minimum of quadratic and rational expression

Question

If $[\begin{array}{ccc} x & 4 & 1 \end{array}] [\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 4 \end{array}] [\begin{array}{c} x \\ 4 \\ - 1 \end{array}] = 0$ then x is equal to

$8 \pm \sqrt{5}$

$- 2 \pm \sqrt{10}$

$3 \pm \sqrt{6}$

$- 1 + \sqrt{6}$

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{[\begin{array}{ccc} x & 4 & 1 \end{array}]}_{1 \times 3} {[\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 4 \end{array}]}_{3 \times 3} {[\begin{array}{c} x \\ 4 \\ - 1 \end{array}]}_{3 \times 1} = 0

{[\begin{array}{ccc} x & 4 & 1 \end{array}]}_{1 \times 3} {[\begin{array}{c} 2 x + 4 + 0 \\ x + 0 - 2 \\ 0 + 8 - 4 \end{array}]}_{3 \times 1} = 0

⇒ 2x² + 4x + 4x – 8 – 4 = 0

⇒ 2x2 + 8x – 12 = 0

⇒ x² + 4x – 6 = 0

⇒ x2 + 2 × 2x + 4 – 4 – 6 = 0

⇒ (x + 2)² = 10

\Rightarrow x + 2 = \pm \sqrt{10}

∴ x = - 2 \pm \sqrt{10}