Problem 8 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A determinant is chosen at random. The set of all determinants of order 2 with elements 0 or 1 only. The probability that value of the determinant chosen is positive is (a) \(3 / 16\) (b) \(3 / 8\) (c) \(1 / 4\) (d) None

Step-by-Step Solution

Answer

The probability is \( \frac{3}{16} \).

Step 1: Understanding the Problem

We are given a set of 2x2 determinants with elements only being 0 or 1. We need to find the probability that a randomly chosen determinant from this set has a positive value.

Step 2: Calculating Total Number of Determinants

Each element of a 2x2 matrix can independently be 0 or 1. Since the matrix has 4 elements, the total number of possible matrices is given by \(2^4 = 16\).

Step 3: Identifying Determinant Formula

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). The determinant is positive if \( ad - bc > 0 \).

Step 4: Listing Configurations with Positive Determinants

For positives: 1. \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \Rightarrow 1*1 - 0*0 = 1 \)2. \( \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \Rightarrow 1*1 - 0*1 = 1 \)3. \( \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \Rightarrow 1*1 - 1*0 = 1 \).

Step 5: Calculating Probability of Positive Determinants

We have identified 3 matrices out of 16 that result in a positive determinant. Thus, the probability is \( \frac{3}{16} \).

Step 6: Conclusion

The probability that the determinant of a randomly chosen 2x2 matrix (elements being 0 or 1) is positive is \( \frac{3}{16} \), which matches answer choice (a).

Key Concepts

Determinant

The determinant is a special number calculated from a square matrix, and it plays an essential role in linear algebra.
When dealing with a small matrix, especially a 2x2 matrix, finding the determinant is quite simple. For a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula to compute the determinant is \( ad - bc \). Essentially, you multiply the top left element \(a\) with the bottom right element \(d\), then subtract the product of the top right element \(b\) and the bottom left element \(c\).

• The determinant can provide insights into the matrix, such as determining whether a matrix is invertible.
• If the determinant is non-zero, the matrix has an inverse; if it's zero, it doesn't.
• Determinants are vital to solving systems of linear equations and can indicate if solutions are unique, none, or infinite.

Understanding determinants allows us to analyze matrices beyond just numbers in an array.

2x2 Matrix

A 2x2 matrix is made up of exactly two rows and two columns. It is essentially a grid consisting of four elements arranged in a square form.
It's structured as follows:

• The matrix is denoted as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
• Each position in the matrix can be a unique value, such as a number, variable, or more complex expression.
• In problems related to 2x2 matrices, the elements can often be specific integers, like in this case, 0 or 1.
• Since there are two elements per row, calculations and manipulations, such as the determinant, can be more straightforward.

This ability to pragmatically handle matrices with simple operations, such as finding determinants and performing other calculations, makes 2x2 matrices particularly useful in teaching foundational concepts in linear algebra.

Matrix Elements

Matrix elements are simply the individual items that comprise the matrix. In a 2x2 matrix, you have exactly four such elements.
Each element can take a specific value depending on the problem at hand.

• In our problem, each element can be either 0 or 1. This means that the matrix can take on different configurations.
• Since there are four elements in a 2x2 matrix, each having two possible values (0 or 1), there are \(2^4 = 16\) possible matrices.
• Understanding and identifying the role and position of each element within the matrix are crucial to computations like the determinant.

These elements' configuration determines the various properties of the matrix; for instance, if specific configurations result in a positive determinant, it helps in answering probability-related questions about determinants.