Question

Assertion :Consider the system of equations$x-2y+3z=-1$ $-x+y-2z=k$ $x-3y+4z=1$

The system of equations has no solution for $k\ne 3,$ and

Reason: The determinant $\left|\begin{array}{ccc}1& 3& -1\\ -1& -2& k\\ 1& 4& 1\end{array}\right|\ne 0,$ for $k\ne 3.$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

The given system of equations can be expressed as

$\left[\begin{array}{ccc}1& -2& 3\\ 1& -3& 4\\ -1& 1& -2\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-1\\ 1\\ k\end{array}\right]$
Applying $R_1\to R_2-R_1,R_3\to R_3+R_1$
$\Rightarrow \left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 1\\ 0& -1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ k-1\end{array}\right]$

Applying $R_3\to R_3-R_2$

$\Rightarrow \left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 1\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ k-3\end{array}\right]$
When $k\ne 3,$ the given system of equations has solution.
$\Rightarrow$ Assertion is true. Clearly, Reason is also true as is rearrangement
of rows and columns of $\left[\begin{array}{ccc}1& -2& 3\\ 1& -3& 4\\ -1& 1& -2\end{array}\right]$
Hence, option (A) is Correct.