The values of $m, n$, for which the system of equations
$$\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$$
has infinitely many solutions, satisfy the equation :
Solution
<p>The given system of linear equations can be represented as,</p>
<p>$$\begin{aligned}
& \left(\begin{array}{ccc|c}
1 & 1 & 1 & 4 \\
2 & 5 & 5 & 17 \\
1 & 2 & m & n
\end{array}\right) \\
& \sim\left(\begin{array}{ccc|c}
1 & 1 & 1 & 4 \\
0 & 3 & 3 & 9 \\
0 & 1 & m-1 & n-4
\end{array}\right) \\
& \sim\left(\begin{array}{ccc|c}
1 & 1 & 1 & 4 \\
0 & 1 & 1 & 3 \\
0 & 0 & m-2 & n-7
\end{array}\right)
\end{aligned}$$</p>
<p>$\because$ System of equations has infinitely many solutions</p>
<p>$\therefore m=2 \& n=7$</p>
<p>Which satisfy equation given in option (1).</p>
<p>$\text { (i.e., } 2^2+7^2-14=39 \text { ) }$</p>
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
This question is part of PrepWiser's free JEE Main question bank. 274 more solved questions on Matrices and Determinants are available — start with the harder ones if your accuracy is >70%.