Consider the system of linear equations $x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$ where $\lambda, \mu \in \mathbf{R}$. Which one of the following statements is NOT correct ?
Solution
<p>$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda{ }^2 z=\mu^2+15$,</p>
<p>$$\Delta=\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & 2 \lambda \\
1 & 3 & 4 \lambda^2
\end{array}\right|=(2 \lambda-1)^2$$</p>
<p>For unique solution $\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}{2}\right)$</p>
<p>Let $\Delta=0, \lambda=\frac{1}{2}$</p>
<p>$$\begin{aligned}
& \Delta_y=0, \Delta_x=\Delta_z=\left|\begin{array}{ccc}
4 \mu & 1 & 1 \\
10 \mu & 2 & 1 \\
\mu^2+15 & 3 & 1
\end{array}\right| \\
& =(\mu-15)(\mu-1)
\end{aligned}$$</p>
<p>For infinite solution $\lambda=\frac{1}{2}, \mu=1$ or 15</p>
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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