If the system of equations
kx + y + 2z = 1
3x $-$ y $-$ 2z = 2
$-$2x $-$2y $-$4z = 3
has infinitely many solutions, then k is equal to __________.
Answer (integer)
21
Solution
D = 0<br><br>$$ \Rightarrow \left| {\matrix{
k & 1 & 2 \cr
3 & { - 1} & { - 2} \cr
{ - 2} & { - 2} & { - 4} \cr
} } \right| = 0$$<br><br>$\Rightarrow$ k (4 $-$ 4) $-$ 1 ($-$ 12 $-$ 4) + 2 ($-$ 6 $-$ 2)<br><br>$\Rightarrow$ 16 $-$ 16 = 0<br><br>Also, ${D_1} = {D_2} = {D_3} = 0$<br><br>$$ \Rightarrow {D_2} = \left| {\matrix{
k & 1 & 2 \cr
3 & 2 & { - 2} \cr
{ - 2} & 3 & { - 4} \cr
} } \right| = 0$$<br><br>$\Rightarrow$ k($-$8 + 6) $-$ 1($-$ 12 $-$ 4) + 2(9 + 4) = 0<br><br>$\Rightarrow$ $-$ 2k + 16 + 26 = 0<br><br>$\Rightarrow$ 2k = 42<br><br>$\Rightarrow$ k = 21
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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