# Understand the problem

The value of \(\lambda\) for which the system of linear equations 2x – y – z = 12 , x -2y +z = -4 , x +y + \(\lambda\)z = 4 has no solution is

##### Source of the problem

Sample Questions (MMA) :2019

##### Topic

Linear Algebra

##### Difficulty Level

Medium

##### Suggested Book

Schaums Outline of Linear Algebra

# Start with hints

Do you really need a hint? Try it first!

Write down the case in the matrix form & try to use the knowledge of system of linear cases 2x -y -z =12, x -2y+ z =-4,x + y + \(\lambda\) z =4

The corresponding matrix format that you will get is The system has no solution if \begin{pmatrix} 12\\-4\\4\\ \end{pmatrix} does not belong to the Sp { \begin{pmatrix} 2 \\ 1\\ 1 \end{pmatrix},\begin{pmatrix} -1 \\ -2\\1 \end{pmatrix},\begin{pmatrix} -1 \\ 1\\\lambda \end{pmatrix}} ———–(1) So, what is the possible value of \(\lambda\)? Can you think the condition (1) in some other ways say in terms of determinants?

If the determinant of matrix

is zero then it has no solution

det A = 0 => \(\lambda\) = -2

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