Difficult
Examine the consistency of the system of equations: $3x - y - 2z = 2$; $2y - z = -1$; $3x - 5y = 3$.
(A) The given system of equations is:
$3x - y - 2z = 2$
$2y - z = -1$
$3x - 5y = 3$
This system can be written as $AX = B$,where
$A = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix}$,$X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$,and $B = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$.
First,we calculate the determinant of $A$:
$|A| = 3(0 - 5) - (-1)(0 - (-3)) + (-2)(0 - 6)$
$|A| = 3(-5) + 1(-3) - 2(-6) = -15 - 3 + 12 = -6$.
Since $|A| \neq 0$,the system is consistent and has a unique solution.
$3x - y - 2z = 2$
$2y - z = -1$
$3x - 5y = 3$
This system can be written as $AX = B$,where
$A = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix}$,$X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$,and $B = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$.
First,we calculate the determinant of $A$:
$|A| = 3(0 - 5) - (-1)(0 - (-3)) + (-2)(0 - 6)$
$|A| = 3(-5) + 1(-3) - 2(-6) = -15 - 3 + 12 = -6$.
Since $|A| \neq 0$,the system is consistent and has a unique solution.
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