System of linear equations

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If A and B are the two real values of $k$ for which the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = k, x + 5 y + 10 z = k^{2}$ is consistent, then $Α + B =$

Moderate

Solution

Augmented matrix $[A B] = [\begin{matrix} 1 & 2 & 1 & 1 \\ 1 & 3 & 4 & k \\ 1 & 5 & 10 & 10^{2} \end{matrix}]$
$_{R_{3} - R_{1}}^{R_{2} - R_{1}} ~ [\begin{matrix} 1 & 2 & 1 & 1 \\ 0 & 1 & 3 & k - 1 \\ 0 & 3 & 9 & k^{2} - 1 \end{matrix}]$
$_{}^{R_{3} - 3 R_{2} ~} [\begin{matrix} 1 & 2 & 1 & 1 \\ 0 & 1 & 3 & k - 1 \\ 0 & 0 & 0 & k^{2} - 3 k + 2 \end{matrix}]$
Since the system is consistent, we have Rank $Α =$ Rank $[A B]$
$k^{2} - 3 k + 2 = 0 \Rightarrow k = 1, 2$
$Α + B = 1 + 2 = 3$

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System of linear equations

Remember concepts with our Masterclasses.

If A and B are the two real values of kfor which the system of equations x+2y+z=1, x+3y+4z=k, x+5y+10z=k2 is consistent, thenΑ+B=

Augmented matrix [AB]=1211134k1510102 R3−R1R2−R1~1211013k-1039k2-1 R3−3R2~ 1211013k-1000k2-3k+2 Since the system is consistent, we have Rank Α=Rank [AB]k2−3 k+2=0 ⇒k=1, 2Α+B= 1+2=3

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If A and B are the two real values of $k$ for which the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = k, x + 5 y + 10 z = k^{2}$ is consistent, then $Α + B =$