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I have the following problem,

Determine how $x_1(\alpha)$ depends on $\alpha$ when $x_1(\alpha)$ is the first component of the solution of the system $Ax = b$, where

A = $\begin{bmatrix}2 & 1 & 1\\1 & -1 & 4\\1 & 1 & -2\end{bmatrix}$ and B = $\begin{bmatrix}-2\\\alpha\\1\end{bmatrix}$

I'm think Cramer's rule but am not sure what that would prove.

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Just using Cramer's rule directly, we get \begin{align} x_1(\alpha)=\frac{\det\begin{pmatrix} \color{blue}{-2}&1&1\\\color{blue}{\alpha}&-1&4\\\color{blue}{1}&1&-2 \end{pmatrix}}{\det A}=\frac{3\alpha+9}{12}=\frac{\alpha+3}{4}. \end{align}

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