Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

A determinant of second order is made with the elements 0 and \(1 .\) The number of determinants with non-negative values is (A) 3 (B) 10 (C) 11 (D) 13

Step-by-Step Solution

Answer

(C) 11

Step 1: Understand the Determinant of a Second Order Matrix

A second order determinant for a 2x2 matrix with elements \( a, b, c, d \) is given by \( \text{det}(A) = ad - bc \). We need to find matrices where \( ad - bc \geq 0 \) using the elements 0 and 1.

Step 2: List All Possible 2x2 Matrices

Each element of the matrix can be either 0 or 1. Therefore, there are \( 2^4 = 16 \) possible matrices since each element \( a, b, c, d \) has 2 options (0 or 1).

Step 3: Calculate Determinants

Evaluate the determinant for each matrix by calculating \( ad - bc \). Enumerate matrices and find the determinant for each one, checking where \( ad - bc \geq 0 \).

Step 4: Identify Matrices with Non-Negative Determinants

Go through each matrix and check if the determinant is non-negative. Count the number of such matrices.

Step 5: Count the Eligible Determinants

After listing all 16 matrices, the ones with non-negative determinants include configurations like squares of zeroes on the diagonal, squares of ones anywhere, rectangles with two ones in diagonal or elsewhere, etc. There should be a total of 11 matrices meeting the condition.

Key Concepts

Second Order Determinants

In linear algebra, a second order determinant involves a 2x2 matrix. This is the simplest form of a determinant, dealing with two rows and two columns.
For a matrix represented as:\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]We calculate the determinant using the formula: \[ \text{det}(A) = ad - bc \]This outcome provides a single number that offers useful properties such as indicating if a matrix is invertible. A determinant of zero suggests the matrix cannot be inverted, while a non-zero determinant indicates it can.
This is a basic tool in matrix algebra, relevant in understanding matrix characteristics and transformations.
When the elements are limited to 0 and 1, as in our exercise, the calculus gets uniquely interesting, leading to specific possible determinations.

Non-Negative Determinants

A determinant is considered non-negative when its calculated value is zero or greater. For our second order matrices, we focus on the condition: \[ ad - bc \geq 0 \]This implies cases where the multiplication of elements on the diagonal (\(ad\)) is no less than the multiplication of elements off the diagonal (\(bc\)).
Non-negative determinants are significant as they indicate certain uniform properties of matrices.

• Determinants of zero imply the linear system has no unique solutions.
• Positive determinants reveal the system's potential solutions or properties like volumes in transformations.

Our task involves identifying matrices where this condition holds true with elements restricted to 0 and 1. Through careful enumeration, we find a variety of combinations where the non-negative condition is naturally satisfied.
Hence, understanding which matrices lead to non-negative determinants is key in matrix evaluation and practical applications of linear transformations.

2x2 Matrices

A 2x2 matrix is a simple, yet crucial structure in the study of linear algebra. They are comprised of 4 elements, usually in the form:\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]This small matrix size makes them perfect for foundational exercises in understanding matrix operations and properties.
Each element in our matrix exercise can be 0 or 1, leading to a total of \(2^4 = 16\) possible configurations due to the binary choice per element.

• Diagonal matrices have either squares of 0s or 1s.
• Mixed diagonal and off-diagonal setup contributes to the determinant values.

Given the limits, evaluating each set thoroughly helps determine which carry the non-negative determinants.
Such matrices, despite their simplicity, form the basis for understanding complex concepts, facilitating deeper insights into larger matrices and more advanced algebraic functions.