Question

The number of solutions of the system of equations
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
is

विकल्प

• 3

• 2

• 1

• 0

Answer 1

$$(d)0$$
The given system of equations can be written in matrix form as follows:
$$\left[\begin{array}{ccc}2& 1& -1\\ 1& -3& 2\\ 1& 4& -3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}7\\ 1\\ 5\end{array}\right]$$
$$AX=B$$
Here,
$$A=\left[\begin{array}{ccc}2& 1& -1\\ 1& -3& 2\\ 1& 4& -3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\text{ and }B=\left[\begin{array}{c}7\\ 1\\ 5\end{array}\right]$$
Now,
$$\left|A\right|=2(9-8)-1(-3-2)-1(4+3)$$
$$=2+5-7$$
$$=0$$
$${\text{ Let }C}_{ij}{\text{ be the cofactors of the elements a }}_{ij}\text{ in }A=\left[a_{ij}\right].\text{ Then,}$$
$$C_{11}={(-1)}^{1+1}\left|\begin{array}{cc}-3& 2\\ 4& -3\end{array}\right|=1,C_{12}={(-1)}^{1+2}\left|\begin{array}{cc}1& 2\\ 1& -3\end{array}\right|=5,C_{13}={(-1)}^{1+3}\left|\begin{array}{cc}1& -3\\ 1& 4\end{array}\right|=7$$
$$C_{21}={(-1)}^{2+1}\left|\begin{array}{cc}1& -1\\ 4& -3\end{array}\right|=-1,C_{22}={(-1)}^{2+2}\left|\begin{array}{cc}2& -1\\ 1& -3\end{array}\right|=-5,C_{23}={(-1)}^{2+3}\left|\begin{array}{cc}2& 1\\ 1& 4\end{array}\right|=-7$$
$$C_{31}={(-1)}^{3+1}\left|\begin{array}{cc}1& -1\\ -3& 2\end{array}\right|=-1,C_{32}={(-1)}^{3+2}\left|\begin{array}{cc}2& -1\\ 1& 2\end{array}\right|=-5,C_{33}={(-1)}^{3+3}\left|\begin{array}{cc}2& 1\\ 1& -3\end{array}\right|=-7$$
$$adjA={\left[\begin{array}{ccc}1& 5& 7\\ -1& -5& -7\\ -1& -5& -7\end{array}\right]}^T$$
$$=\left[\begin{array}{ccc}1& -1& -1\\ 5& -5& -5\\ 7& -7& -7\end{array}\right]$$
$$\Rightarrow \left(adjA\right)B=\left[\begin{array}{ccc}1& -1& -1\\ 5& -5& -5\\ 7& -7& -7\end{array}\right]\left[\begin{array}{c}7\\ 1\\ 5\end{array}\right]$$
$$=\left[\begin{array}{c}7-1-5\\ 35-5-25\\ 49-7-35\end{array}\right]$$
$$=\left[\begin{array}{c}1\\ 5\\ 7\end{array}\right]\ne 0$$
So, the given system of equations has no solution.