The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :
Solution
$$\Delta = \left| {\matrix{
3 & { - 2} & { - k} \cr
1 & { - 4} & { - 2} \cr
1 & 2 & { - 1} \cr
} } \right| = 0$$<br><br>$3(4 + 4) + 2( - 2 + 2) - k(4 + 4) = 0$<br><br>$\Rightarrow k = 3$<br><br>$${\Delta _x} = \left| {\matrix{
{10} & { - 2} & { - 3} \cr
6 & { - 4} & { - 2} \cr
{5m} & 2 & { - 1} \cr
} } \right| \ne 0$$<br><br>$10(4 + 4) + 2( - 6 + 10m) - 3(12 + 20m) \ne 0$<br><br>$80 - 12 + 20m - 36 - 60m \ne 0$<br><br>$40m \ne 32 \Rightarrow m \ne {4 \over 5}$<br><br>$${\Delta _y} = \left| {\matrix{
3 & {10} & { - 3} \cr
2 & 6 & { - 2} \cr
1 & {5m} & { - 1} \cr
} } \right| \ne 0$$<br><br>$3( - 6 + 10m) - 10( - 2 + 2) - 3(10m - 6) \ne 0$<br><br>$- 18 + 30m - 30m + 18 \ne 0 \Rightarrow 0$<br><br>$${\Delta _z} = \left| {\matrix{
3 & { - 2} & {10} \cr
2 & { - 4} & 6 \cr
1 & 2 & {5m} \cr
} } \right| \ne 0$$<br><br>$$3( - 20m - 12) + 2(10m - 6) + 10(4 + 4) - 40m + 32 \ne 0 \Rightarrow m \ne {4 \over 5}$$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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