Problem 11 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The least value of the product \(x y z\) for which the determinant \(\left|\begin{array}{ccc}x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z\end{array}\right|\) is non-negative, is: [Online April 10, 2015] (a) \(-2 \sqrt{2}\) (b) \(-1\) (c) \(-16 \sqrt{2}\) (d) \(-8\)

Step-by-Step Solution

Answer

The least value of the product \(xyz\) is \(-8\).

Step 1: Understand the Problem

We are given a determinant of a 3x3 matrix \( \begin{pmatrix} x & 1 & 1 \ 1 & y & 1 \ 1 & 1 & z \end{pmatrix} \). We need to find the least value of the product \( xyz \) such that the determinant is non-negative.

Step 2: Write the Determinant Expression

Calculate the determinant of the matrix:\[ \left| \begin{array}{ccc} x & 1 & 1 \ 1 & y & 1 \ 1 & 1 & z \end{array} \right| = x(yz - 1) - 1(z - 1) + 1(1-y) \].

Step 3: Simplify the Determinant Formula

Substitute the expansions in place in the determinant formula:\[ = xyz - x - z + 1 + 1 - y \]\[ = xyz - x - y - z + 2 \].

Step 4: Set Inequality for Non-Negativity

Set the equation to be greater than or equal to zero for the determinant to remain non-negative:\[ xyz - x - y - z + 2 \geq 0 \].

Step 5: Analyze Via Substitution

Try substituting small integer values for \(x, y,\) and \(z\) that minimize the product \(xyz\), ensuring that the inequality is satisfied. Testing \(x = y = z = -2\) makes the product \(xyz = -2 \times -2 \times -2 = -8\) and satisfies the inequality since\[ -8 - (-2) - (-2) - (-2) + 2 = 0 \] which is non-negative.

Key Concepts

3x3 Matrix

A 3x3 matrix is a simple concept from linear algebra. It consists of three rows and three columns.This gives nine elements in total. Mathematicians and scientists often use these matrices to solve linear equations and model real-world phenomena.

In this case, we are working with a specific 3x3 matrix:

• First row: \( x, 1, 1 \)
• Second row: \( 1, y, 1 \)
• Third row: \( 1, 1, z \)

Each element is denoted by a letter, representing a variable or a constant.

To solve problems involving matrices, it's essential to become familiar with this type of arrangement and practice calculating their determinants.

Non-negative Determinant

A determinant is a special number calculated from a square matrix. It reflects certain properties of the matrix, such as invertibility.When we talk about a non-negative determinant, we mean that the determinant value is either zero or a positive number.

For matrices, non-negative determinants are significant because:

• They imply the system of equations represented by the matrix has solutions.
• They help determine if a matrix is invertible or singular (non-invertible).

In our exercise, we aim to find the least value of the product \( x y z \) such that the determinant remains non-negative.This involves solving inequalities and substituting values, making sure that the determinant doesn't become negative.

Matrix Algebra

Matrix algebra deals with operations on matrices, including addition, multiplication, and calculating determinants.These operations play a crucial role in various fields, from computer science to engineering.

To find the least product \( x y z \) in the given challenge, the steps involve:

• Understanding and computing the determinant of the matrix.
• Ensuring that the result of operations like multiplication and subtraction maintains the required conditions (e.g., non-negativity).

Matrix algebra often requires breaking down complex problems into manageable parts, like calculating determinants for sub-matrices to piece together the whole picture.

Product Minimization

Minimizing a product like \( x y z \) involves finding the smallest possible value subject to given constraints.These constraints are typically derived from algebraic conditions, as in our example with the non-negative determinant.

The key steps for product minimization in this context are:

• Establishing an inequality based on the determinant condition.
• Substituting values thoughtfully to test which combinations yield the smallest acceptable product.

In our exercise, finding \( x = y = z = -2 \) was a successful approach.This solution satisfied the condition \( xyz - x - y - z + 2 \geq 0 \) while giving the minimum product of \(-8\).