For real numbers $\alpha$ and $\beta$, consider the following system of linear equations :
x + y $-$ z = 2, x + 2y + $\alpha$z = 1, 2x $-$ y + z = $\beta$. If the system has infinite solutions, then $\alpha$ + $\beta$ is equal to ______________.
Answer (integer)
5
Solution
For infinite solutions<br><br>$\Delta$ = $\Delta$<sub>1</sub> = $\Delta$<sub>2</sub> = $\Delta$<sub>3</sub> = 0<br><br>$\Delta$ = $$\left| {\matrix{
1 & 1 & { - 1} \cr
1 & 2 & \alpha \cr
2 & { - 1} & 1 \cr
} } \right| = 0$$<br><br>$$\Delta = \left| {\matrix{
3 & 0 & 0 \cr
1 & 2 & \alpha \cr
2 & { - 1} & 1 \cr
} } \right| = 0$$<br><br>$\Delta$ = 3(2 + $\alpha$) = 0<br><br>$\Rightarrow$ $\alpha$ = $-$2<br><br>$${\Delta _2} = \left| {\matrix{
1 & 2 & { - 1} \cr
1 & 1 & { - 2} \cr
2 & \beta & 1 \cr
} } \right| = 0$$<br><br>1(1 + 2$\beta$) $-$2(1 + 4) $-$ ($\beta$ $-$ 2) = 0<br><br> $\beta$ $-$ 7 = 0<br><br>$\beta$ = 7<br><br>$\therefore$ $\alpha$ + $\beta$ = 5
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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