Problem 21 - A Complete Resource Book in Mathematics for JEE Main 2017

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: Understanding Determinant of Second Order

A second order determinant is a 2x2 matrix, and its determinant is calculated as \( ad - bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix: \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).

Step 2: Counting Possible Matrices

We are given that each element of the matrix can be either \( 0 \) or \( 1 \). Since there are four positions in the matrix, there are \( 2^4 = 16 \) possible matrices.

Step 3: Identifying Matrices with Non-Negative Determinants

The determinant \( ad - bc \) must be non-negative. Hence, we want \( ad \geq bc \). If both \( a \) and \( d \) are \( 0 \), or both \( b \) and \( c \) are \( 0 \), \( ad - bc \geq 0 \). Also, consider cases where \( ad \) results in a higher or equal value compared to \( bc \).

Step 4: Counting Valid Determinants

List all valid combinations of \( a, b, c, \) and \( d \):1. \( a, b, c, d = 0 \) (1 case)2. One \( 1 \): \( a, d, bc = 1,0 \, \text{or} \, 0,1 \) (3 cases)3. Two 1s: \( ad \text{ and } bc = 0, \, 1 \, \text{or} \, 1,0 \) (6 cases, except where both \( a=1, d=1 \) with neither \( b \) nor \( c \) being 1)4. Three 1s: \( a, b, c, or d eq 0 \) (3 cases)Summing these gives 11 valid determinants.

Key Concepts

Second Order Determinant

When we talk about a second order determinant, we're discussing a concept that comes from a 2x2 matrix. This kind of matrix has two rows and two columns, and it looks like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Where \(a, b, c,\) and \(d\) are the elements of the matrix. The "determinant" of this matrix is a special number that you calculate using the formula: \[ad - bc\]What this formula means is that you multiply the elements \(a\) and \(d\), then subtract the product of \(b\) and \(c\). The determinant is a very useful value that gives us important information about the matrix. It can tell us, for example, whether the matrix has an inverse or not. In our specific exercise, we are looking at determinants that only use the numbers 0 and 1.

Matrix Elements

Matrix elements are the individual numbers that make up a matrix. For a 2x2 matrix, these elements are specifically the numbers \(a, b, c,\) and \(d\) as shown in the matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]In our exercise, the possible values for these elements are limited to just 0 and 1. This makes our calculations straightforward, but also requires us to think about all possible combinations. Since there are four elements in the matrix and each can independently be 0 or 1, we have:

• 2 choices for \(a\),
• 2 choices for \(b\),
• 2 choices for \(c\),
• 2 choices for \(d\).

When you multiply these choices together, you get \(2^4 = 16\) total possible matrices. The combinations of these elements determine the value of the determinant.

Non-negative Determinant Values

In this exercise, we are specifically looking for the matrices that have non-negative determinant values. A determinant is non-negative if it is zero or a positive number. For our matrix determinant \(ad - bc\), it means:

• \(ad\) must be greater than or equal to \(bc\).

Let's break it down:- If both \(a\) and \(d\) are zero, then \(ad\) is automatically zero, ensuring \(ad - bc \geq 0\), regardless of \(b\) and \(c\), as long as they are also not both 1.- If one of \(a\) or \(d\) is 1, the product \(ad\) is 1, which can only be non-negative if \(bc\) (product of \(b\) and \(c\)) is zero.- In situations where the number of 1's is three (one element is zero), it's usually possible to align the arrangement to ensure \(ad \geq bc\).By taking into account all these logical possibilities, and listing cases systematically, we find that 11 such determinations do indeed meet this non-negative condition.