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સમીકરણોની સંહિતાની સુસંગતતા ચકાસો : $3 x-y-2 z=2$ ; $2 y-z=-1$ ; $3 x-5 y=3$
Solution
The given system of equation is:
$3 x-y-2 z=2$
$2 y-z=-1$
$3 x-5 y=3$
This system of equations can be written in the form of $A X=B$, where
$A=\left[\begin{array}{ccc}
3 & -1 & -2 \\
0 & 2 & -1 \\
3 & -5 & 0
\end{array}\right], X=\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right] \text { and } B=\left[\begin{array}{c}
2 \\
-1 \\
3
\end{array}\right]$
Now,
$|A|=3(-5)-0+3(1+4)=-15+15=0$
$\therefore$ $A$ is a singular matrix.
Now,
$(a d j A)=\left[\begin{array}{lcl}-5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6\end{array}\right]$
$\therefore (adjA)B = \left[ {\begin{array}{*{20}{c}}
{ – 5}&{10}&5 \\
{ – 3}&6&3 \\
{ – 6}&{12}&6
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
2 \\
{ – 1} \\
3
\end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}}
{ – 10 – 10 + 15} \\
{ – 6 – 6 + 9} \\
{ – 12 – 12 + 18}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{ – 5} \\
{ – 3} \\
{ – 6}
\end{array}} \right] \ne 0$
Thus, the solution of the given system of equation does not exist. Hence, the system of equations is inconsistent.
