First slide

System of linear equations

Question

The system of equations $- 2 x + y + z = a, x - 2 y + z = b, x + y - 2 z = c$ is inconsistent, if

Moderate

A

$a + b + c = 0$
B

$a + b + c \leq 0$
C

$a + b + c \neq 0$
D

$a + b + c \geq 0$

Solution

$W e h a v e Δ = |\begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix}| = - 2 (4 - 1) - 1 (- 2 - 1) + 1 (1 + 2) = 0$
The system of equation is inconsistent if $|\begin{array}{l} - 2 1 - a \\ 1 - 2 - b \\ 1 1 - c \end{array}| \neq 0$
$\begin{array}{l} \Rightarrow - 2 [2 c + b] - 1 [- c + b] - a (3) \neq 0 \\ \Rightarrow - 4 c - 2 b + c - b - 3 a \neq 0 \\ \Rightarrow - 3 [a + b + c] \neq 0 \Rightarrow a + b + c \neq 0 \end{array}$

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