If , then find X.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Multiplication of Matrices

Question

If $[\begin{array}{ccc} 1 & x & 1 \end{array}] [\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{array}] [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}] = 0$ , then find X.

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Given $[\begin{array}{ccc} 1 & x & 1 \end{array}] [\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{array}] [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}] = 0$

\Rightarrow [\begin{array}{ccc} 1 \times 1 + x \times 4 + 1 \times 3 & 1 \times 2 + 5 x + 1 \times 2 & 3 \times 1 + 6 x + 1 \times 5 \end{array}] [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}] = 0

\Rightarrow [\begin{array}{ccc} 1 + 4 x + 3 & 2 + 5 x + 2 & 3 + 6 x + 5 \end{array}] [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}] = 0

\Rightarrow [\begin{array}{ccc} 4 x + 4 & 5 x + 4 & 6 x + 8 \end{array}] [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}] = 0

⇒ (4x + 4) × 1 + (5x + 4) × 2 + (6x + 8) × 3 = 0

⇒ 4x + 4 + 10x + 8 + 18x + 24 = 0

⇒ 32x + 36 = 0 ⇒ 32x = – 36

\Rightarrow x = \frac{- 9}{8}