Problem 117 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the system of equations \(\quad\) [Online May 7, 20 \(x+y+z=6\) \(x+2 y+3 z=10\) \(x+2 y+\lambda z=0\) has a unique solution, then \(\lambda\) is not equal to (a) 1 (b) 0 (c) 2 (d) 3

Step-by-Step Solution

Answer

Lambda is not equal to 3.

Step 1: Set Up the Matrix

Write the system of equations in matrix form as follows:\[\begin{bmatrix} 1 & 1 & 1 \1 & 2 & 3 \1 & 2 & \lambda \end{bmatrix} \begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 6 \10 \0 \end{bmatrix}\]

Step 2: Calculate the Determinant

To determine if the system has a unique solution, ensure that the determinant of the coefficient matrix is non-zero. Find the determinant of:\[\begin{vmatrix} 1 & 1 & 1 \1 & 2 & 3 \1 & 2 & \lambda \end{vmatrix}\]

Step 3: Evaluate the Determinant

Calculate the determinant using the formula for a 3x3 matrix, \[\text{det} = 1(2\lambda - 3\times2) - 1(1\lambda - 3\times1) + 1(1\times2 - 2\times1)\]This becomes \[2\lambda - 6 - (\lambda - 3) + (2 - 2)\].

Step 4: Simplify the Determinant

Simplify the equation obtained from the determinant:\[2\lambda - 6 - \lambda + 3\].Which simplifies to \[\lambda - 3\].

Step 5: Set Determinant Not Equal to Zero

For the system to have a unique solution, the determinant must not be equal to zero:\[\lambda - 3 eq 0\].

Step 6: Solve the Inequality

Solving the inequality \[\lambda - 3 eq 0\] gives \[\lambda eq 3\]. This means \(\lambda\) cannot be 3 for a unique solution.

Key Concepts

Matrices

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is a powerful tool for encapsulating information and solving systems of equations. When working with a system of linear equations, like the problem we have, matrices provide a very structured way to organize the coefficients and constants involved. In matrix form, an equation system is typically written as:

• A coefficient matrix that holds the variables' coefficients.
• A column matrix (or vector) for the variables.
• A constant matrix (or vector) for the solutions.

For example, our system of equations has a corresponding matrix representation: \[\begin{bmatrix} 1 & 1 & 1 \1 & 2 & 3 \1 & 2 & \lambda \end{bmatrix} \begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 6 \10 \0 \end{bmatrix}\]Matrices make it easier to perform operations such as adding equations, multiplying them, or finding solutions using methods like Gaussian elimination.

Determinant

The determinant is a special number that can be calculated from the elements of a square matrix. It is a key to understanding many properties of matrices and the systems of equations they represent. In a square matrix, the determinant helps us determine whether the matrix has an inverse. An important property regarding the inverse is crucial for solutions:

• If the determinant is not zero, the matrix has an inverse, indicating that the system of equations has a unique solution.
• If the determinant is zero, the matrix is singular, meaning the system has either no solution or infinitely many solutions.

For a 3x3 matrix, the determinant is found using the formula that involves multiplication and subtraction of its elements. In the problem presented, the calculation: \[\text{det} = 2\lambda - 6 - (\lambda - 3) + (2 - 2)\]ensures that the determinant is not zero to guarantee a unique solution. So, when we simplify, we find that the determinant must not be zero, which occurs when \( \lambda eq 3 \).

Unique Solution

In a system of linear equations, having a unique solution means that there is exactly one set of values for the variables that satisfy all the equations simultaneously. For a system to have a unique solution, two major conditions must be met:

• The number of equations must equal the number of variables.
• The coefficient matrix must have a non-zero determinant, indicating that the matrix is invertible.

Relating to our problem, this means ensuring \( \lambda eq 3 \). When the determinant of our coefficient matrix is non-zero (i.e., when \( \lambda - 3 eq 0 \)), it ensures that the system of equations can be solved uniquely for the variables \( x, y, \) and \( z \). This is crucial because it guarantees that the relationships between variables are so specific that they produce one exact answer instead of many possible solutions or none at all.