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Answer:

Inconsistent

Given:

$3 x-y-2 z=2 \quad, \quad 2 y-z=-1 \quad, \quad 3 x-5 y=3$

Hint:

Inconsistent means two or more equations that are impossible to solve based on using one set of values for variables.

Solution:

The given system of equations can be expressed as follows

\begin{aligned} &A X=B\\ &\text { Here, }\\ &A=\left[\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{array}\right] \quad, \quad X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{c} 2 \\ -1 \\ 3 \end{array}\right] \end{aligned}

\begin{aligned} &\text { Now, }\\ &\begin{aligned} |A|=\left|\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{array}\right| &=3(0-5)+1(0+3)-2(0-6) \\ &=-15+3+12 \\ &=0 \end{aligned} \end{aligned}

$C_{i j}\; be\; the\; co-\! f\! actor\; o\! f\; the\; elements\; a_{i j}\; in \; A=\left[a_{i j}\right].\; Then,$

\begin{aligned} &C_{11}=-1^{1+1}\left|\begin{array}{cc} 2 & -1 \\ -5 & 0 \end{array}\right|=-5 \quad, \quad C_{21}=-1^{2+1}\left|\begin{array}{cc} 0 & -1 \\ 3 & 0 \end{array}\right|=-3 \\ &C_{12}=-1^{1+2}\left|\begin{array}{cc} 0 & 2 \\ 3 & -5 \end{array}\right|=-6 \quad, \quad C_{22}=-1^{2+2}\left|\begin{array}{cc} -1 & -2 \\ -5 & 0 \end{array}\right|=10 \end{aligned}

\begin{aligned} &C_{13}=-1^{1+3}\left|\begin{array}{cc} 3 & -2 \\ 3 & 0 \end{array}\right|=6 \quad, \quad C_{23}=-1^{2+3}\left|\begin{array}{ll} 3 & -1 \\ 3 & -5 \end{array}\right|=12 \\ &C_{31}=-1^{3+1}\left|\begin{array}{ll} -1 & -2 \\ 2 & -1 \end{array}\right|=5 \quad, \quad C_{32}=-1^{3+2}\left|\begin{array}{ll} 3 & -2 \\ 0 & -1 \end{array}\right|=3 \\ &C_{33}=-1^{3+3}\left|\begin{array}{cc} 3 & -1 \\ 0 & 2 \end{array}\right|=6 \end{aligned}

$\begin{gathered} \operatorname{adjA}=\left[\begin{array}{ccc} -5 & -3 & -6 \\ 10 & 6 & 12 \\ 5 & 3 & 6 \end{array}\right]^{T} \\ (\text { adjA }) B=\left[\begin{array}{ccc} -5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6 \end{array}\right]\left[\begin{array}{c} 2 \\ -1 \\ 3 \end{array}\right] \\ =\left[\begin{array}{c} -10-10+15 \\ -6-6+9 \\ -12-12+18 \end{array}\right] \\ =\left[\begin{array}{c} -5 \\ -3 \\ -6 \end{array}\right] \neq 0 \end{gathered}$

Hence, the given system of equation is inconsistent.

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