The values of $\lambda$ and $\mu$ for which the system of linear equations
x + y + z = 2
x + 2y + 3z = 5
x + 3y + $\lambda$z = $\mu$
has infinitely many solutions are, respectively:
Solution
For infinite many solutions
<br><br>D = D<sub>1</sub> = D<sub>2</sub> = D<sub>3</sub> = 0
<br><br>Now D = $$\left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & \lambda \cr
} } \right|$$ = 0
<br><br>$\Rightarrow$ 1. (2$\lambda$ – 9) –1.($\lambda$ – 3) + 1.(3 – 2) = 0
<br><br>$\Rightarrow$ $\lambda$ = 5
<br><br>Now D<sub>1</sub> = $$\left| {\matrix{
2 & 1 & 1 \cr
5 & 2 & 3 \cr
\mu & 3 & 5 \cr
} } \right|$$ = 0
<br><br>$\Rightarrow$ 2(10 – 9) –1(25 – 3$\mu$) + 1(15 – 2$\mu$) = 0
<br><br>$\Rightarrow$ $\mu$ = 8
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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