If , then x =

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Quadratic Equation and Nature of Roots

Question

If $[\begin{array}{ccc} 1 & x & 1 \end{array}] [\begin{array}{ccc} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{array}] [\begin{array}{c} 1 \\ 1 \\ x \end{array}] = 0$ , then x =

$\frac{- 7 \pm \sqrt{53}}{2}$

$\frac{- 7 \pm \sqrt{35}}{2}$

$\frac{- 9 \pm \sqrt{35}}{2}$

$\frac{- 9 \pm \sqrt{53}}{2}$

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Given,

[\begin{array}{ccc} 1 & x & 1 \end{array}] [\begin{array}{ccc} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{array}] [\begin{array}{c} 1 \\ 1 \\ x \end{array}] = 0

Upon multiplying the first 2 matrices, we get,

(\begin{array}{ccc} 1 \cdot 1 + x \cdot 0 + 1 \cdot 0 & 1 \cdot 3 + x \cdot 5 + 1 \cdot 3 & 1 \cdot 2 + x \cdot 1 + 1 \cdot 2 \end{array}) (\begin{array}{c} 1 \\ 1 \\ x \end{array}) = 0

Upon further simplification, we get,

(\begin{array}{ccc} 1 & 6 + 5 x & 4 + x \end{array}) (\begin{array}{c} 1 \\ 1 \\ x \end{array}) = 0

Again multiplying the above matrices, we get,

(1.1 + (6.5x). 1 + (4 + x)x) = 0

Upon further simplification, we get,

(x² + 9x + 7) = 0

Solving the above quadratic equation, we get,

x = \frac{- 9 \pm \sqrt{53}}{2}