For which of the following ordered pairs ($\mu$, $\delta$),
the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = $\mu$
4x + 4y + 4z = $\delta$
is inconsistent ?
Solution
For inconsistent system we need
<br><br>$\Delta$ = 0 and atleast one of $\Delta$x, $\Delta$y, $\Delta$z $\ne$ 0
<br><br>$\therefore$ $\Delta$ = $$\left| {\matrix{
1 & 2 & 3 \cr
3 & 4 & 5 \cr
4 & 4 & 4 \cr
} } \right|$$ = 0
<br><br>$\Delta$<sub>x</sub> = $$\left| {\matrix{
1 & 2 & 3 \cr
\mu & 4 & 5 \cr
\delta & 4 & 4 \cr
} } \right|$$
<br><br>= (-4) - 2($\mu$ - 5$\delta$) + 3(4$\mu$ - 4$\delta$)
<br><br>$\Rightarrow$ 2$\mu$ $\ne$ $\delta$ + 2 ....(1)
<br><br>Only ($\mu$, $\delta$) = (4, 3) does satisfy the equation (1).
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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