For what values of x- -1 2 1- left bmatrix182 - 0 12 - 0 - 11 -1 - 0 - 2bmatrix bmatrix 011211Xendbmatrix -right -0 -

Given, [1 2 1] ( [ 1 amp; 2 amp; 0; 2 amp; 0 amp; 1; 1 amp; 0 amp; 2 ][ 0; 2; x ] )= 0 ⇒ [1 2 1][ 1 amp; 2 amp; 0; 2 amp; 0 amp; 1; ...

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Standard XII

Mathematics

Scalar Multiplication of a Matrix

For what valu...

Question

For what values of x, [1 2 1] $⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 2 & 0 & 1 1 & 0 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 02 x \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠ = 0$ ?

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Solution

Given, [1 2 1] $⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 2 & 0 & 1 1 & 0 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 02 x \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠ = 0$

$\Rightarrow [121] ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 2 & 0 & 1 1 & 0 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 02 x \end{matrix} ⎤ ⎥ ⎦ = 0 \Rightarrow [123] ⎡ ⎢ ⎣ \begin{matrix} 0 + 4 + 0 0 + 0 + x 0 + 0 + 2 x \end{matrix} ⎤ ⎥ ⎦ = 0$
(because Matrix multiplication is associative)
$\Rightarrow [123] ⎡ ⎢ ⎣ \begin{matrix} 4 x 2 x \end{matrix} ⎤ ⎥ ⎦ = 0 \Rightarrow [4 + 2 x + 2 x] = 0 \Rightarrow 4 + 4 x = 0 \Rightarrow x = - 1$ .

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For what values of x, [1 2 1] ⎛⎜⎝⎡⎢⎣120201102⎤⎥⎦⎡⎢⎣02x⎤⎥⎦⎞⎟⎠=0 ?

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