The number of real values of $\lambda$, such that the system of linear equations
2x $-$ 3y + 5z = 9
x + 3y $-$ z = $-$18
3x $-$ y + ($\lambda$2 $-$ | $\lambda$ |)z = 16
has no solutions, is
Solution
<p>$$\Delta = \left| {\matrix{
2 & { - 3} & 5 \cr
1 & 3 & { - 1} \cr
3 & { - 1} & {{\lambda ^2} - |\lambda |} \cr
} } \right| = 2\left( {3{\lambda ^2} - 3|\lambda | - 1} \right) + 3\left( {{\lambda ^2} - |\lambda | + 3} \right) + 5( - 1 - 9)$$</p>
<p>$= 9{\lambda ^2} - 9|\lambda | - 43$</p>
<p>$= 9|\lambda {|^2} - 9|\lambda | - 43$</p>
<p>$\Delta = 0$ for 2 values of $|\lambda |$ out of which one is $-\mathrm{ve}$ and other is $+\mathrm{ve}$</p>
<p>So, 2 values of $\lambda$ satisfy the system of equations to obtain no solution.</p>
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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