SATHEE: Matrices and Determinants 4 Question 33

Matrices and Determinants 4 Question 33

34. For a real number $α$ , if the system

$[\begin{matrix} 1 & α & α^{2} \\ α & 1 & α \\ α^{2} & α & 1 \end{matrix}]$ $[\begin{matrix} x \\ y \\ z \end{matrix}]$ = $[\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$

of linear equations, has infinitely many solutions, then $1 + α + α^{2} =$

(2017 Adv.)

Show Answer

Answer:

Correct Answer: 34. (1)

Solution:

$[\begin{matrix} 1 & α & α^{2} \\ α & 1 & α \\ α^{2} & α & 1 \end{matrix}] = 0$

$\Rightarrow α^{4} - 2 α^{2} + 1 = 0$

$\Rightarrow α^{2} = 1$

$\Rightarrow α = \pm 1$

But $α = 1$ not possible

$∴ α = - 1$

[Not satisfying equation]

Hence, $1 + α + α^{2} = 1$

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Matrices and Determinants 4 Question 33

34. For a real number α, if the system

Answer:

Solution:

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34. For a real number $α$ , if the system