Find the value of x, if

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Find the value of x, if $[\begin{array}{ccc} 1 & x & 1 \end{array}] [\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{array}] [\begin{array}{c} 1 \\ 2 \\ x \end{array}] = 0$

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We have, ${[\begin{array}{ccc} 1 & x & 1 \end{array}]}_{1 \times 3} {[\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{array}]}_{3 \times 3} {[\begin{array}{c} 1 \\ 2 \\ x \end{array}]}_{3 \times 1} = 0$
$\Rightarrow {[\begin{array}{ccc} 1 + 2 x + 15 & 3 + 5 x + 3 & 2 + x + 2 \end{array}]}_{1 \times 3} {[\begin{array}{c} 1 \\ 2 \\ x \end{array}]}_{3 \times 1} = 0$
⇒ [16 + 2x + (5x + 6)⋅2 + (x + 4)x]_1×1 = 0

⇒ [16 + 2x + 10x + 12 + x² + 4x] = 0

⇒ [x² + 16x + 28] = 0

⇒ [x² + 2x + 14x + 28] = 0

⇒ (x + 2)(x + 14) = 0

∴ x = 2, –14

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