# 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
Linear Algebra
Medium
##### Suggested Book
Schaums Outline of Linear Algebra

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 $\begin{pmatrix} 2&-1&-1\\ 1&-2&1\\ 1&1&\lambda\\ \end{pmatrix} \begin{pmatrix} x \\ y\\ z \end{pmatrix}=\begin{pmatrix} 12\\-4\\4\\ \end{pmatrix}$   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 $A=\begin{pmatrix} 2&-1&-1\\ 1&-2&1\\ 1&1&\lambda\\ \end{pmatrix}$

is zero then it has no solution

det A = 0 => $$\lambda$$ = -2

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Similar Problems

## An excursion in Linear Algebra

Did you know Einstein badly needed linear algebra? We will begin from scratch in this open seminar and master useful tools on the way. The open seminar on linear algebra is coming up on 14th November 2019, (8 PM IST).

## Linear Algebra total recall (Open Seminar)

Open Seminar on linear algebra. A review of all major ideas. Even if you have little or no knowledge about Linear Algebra, you may join. Register now.

## 4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 2

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 1

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## Sum based on Probability – ISI MMA 2018 Question 24

This is an interesting and cute sum based on the concept of Arithematic and Geometric series .The problem is to find a solution of a probability sum.

## System of the linear equation: ISI MMA 2018 Question 11

This is a cute and interesting problem based on System of the linear equation in linear algebra. Here we are finding the determinant value .

## Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .

## To find Trace of a given Matrix : ISI MMA 2018 Question 13

These is a cute and interesting sum where the trace of a given matrices needs to be found using a very simple but effective method