GATE 2024 ME – Question 14

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 1 mark · Multiple choice

Consider the system of linear equations

$x + 2y + z = 5$

$2x + ay + 4z = 12$

$2x + 4y + 6z = b$

The values of $a$ and $b$ such that there exists a non-trivial null space and the system admits infinite solutions are

$a = 8$, $b = 14$
$a = 4$, $b = 12$
$a = 8$, $b = 12$
$a = 4$, $b = 14$

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Correct answer: (D) $a = 4$, $b = 14$

Explanation

A non-trivial null space needs a zero determinant: $\begin{vmatrix} 1 & 2 & 1 \\ 2 & a & 4 \\ 2 & 4 & 6 \end{vmatrix} = 1(6a - 16) - 2(12 - 8) + 1(8 - 2a) = 4a - 16 = 0$, so $a = 4$. With $a = 4$ the second equation is $2x + 4y + 4z = 12$ and twice the first is $2x + 4y + 2z = 10$. Subtracting gives $2z = 2$, so $z = 1$. The third equation minus the second gives $2z = b - 12$, which must also give $z = 1$, so $b = 14$.