Problem 2 - Algebra for JEE ADVANCED

Question

If a determinant of order \(3 \times 3\) is formed by using the numbers 1 or \(-1\), then the minimum value of the determinant is a. \(-2\) b. \(-4\) c. 0 d. \(-8\)

Step-by-Step Solution

Answer

The minimum value is -4 (option b).

Step 1: Understanding the Determinant of a 3x3 Matrix

The determinant of a matrix \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \] is given by the formula: \[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg). \] In this scenario, matrix elements can be either 1 or -1.

Step 2: Setup a Matrix and Calculate Its Determinant

Choose one possible matrix for evaluation. For instance, let's take \[ \begin{bmatrix} 1 & 1 & 1 \ -1 & -1 & 1 \ 1 & -1 & -1 \end{bmatrix} \].Calculate its determinant:\[ \text{det} = 1((-1) \times (-1) - 1 \times (-1)) - 1((-1) \times (-1) - 1 \times 1) + 1((-1) \times (-1) - (-1) \times -1) \]\[ = 1(1 + 1) - 1(1 - 1) + 1(1 - 1) \]\[ = 2 - 0 + 0 \]\[ = 2 \].

Step 3: Try More Combinations

Keep trying different combinations of 1 and -1 as elements of the matrix while re-evaluating the determinant each time to find the possible minimum.As an example, take another matrix: \[ \begin{bmatrix} 1 & 1 & 1 \ 1 & -1 & 1 \ 1 & 1 & -1 \end{bmatrix} \] .Calculate the determinant:\[ \text{det} = 1(-1 \times -1 - 1 \times 1) - 1(1 \times -1 - 1 \times 1) + 1(1 \times 1 - (-1) \times 1) \]\[ = 1(1 - 1) - 1(-1 - 1) + 1(1 + 1) \]\[ = 0 + 2 + 2 \]\[ = 4\].

Step 4: Determine the Minimum Possible Value

After several combinations, consider the matrix \[ \begin{bmatrix} 1 & 1 & 1 \ 1 & -1 & -1 \ -1 & -1 & 1 \end{bmatrix} \].Calculate the determinant:\[ \text{det} = 1(-1 \times 1 - (-1)\times(-1)) - 1(1 \times 1 - (-1) \times (-1)) + 1(1 \times (-1) - -1 \times 1) \]\[ = 1(-1 - 1) - 1(1 - 1) + 1(-1 - 1) \]\[ = -2 - 0 - 2 \]\[ = -4 \].Jack finds that this is the minimum determinant value.

Key Concepts

3x3 matrix

A 3x3 matrix is a square array that consists of three rows and three columns. Each position in the array is filled with a numerical value, which can be used in various mathematical computations.
A matrix of this size is essential in linear algebra because it helps in performing linear transformations and solving systems of linear equations, among others. For example, a typical 3x3 matrix can be represented as follows:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]Here, each letter (a, b, c... etc.) stands for the elements or values the matrix can have. With the exercise at hand, we are limited to element values of either 1 or -1. This constraint is part of the exercise's challenge and offers an insight into permutations that affect the determinant.

Matrix elements

Matrix elements are the individual values or numbers within the matrix. In a 3x3 matrix, there are exactly nine elements. Each element's position is significant—it dictates how the element contributes to computations such as determinant calculations. Understanding elements:

• The first element in the first row is usually denoted as 'a', in standard notation.
• The process from left to right and top to bottom continues with labels like b, c, and so on.

In the given exercise, the elements can either be 1 or -1.
This means that each element can drastically change the resultant calculations due to the limited binary options for values. Knowing the role of each element is crucial for determining how each can be manipulated for different outcomes in determinant calculations. This manipulation forms a fundamental part of exploring permutations in matrices.

Determinant calculation

The determinant of a matrix is a special number associated with a square matrix. For a 3x3 matrix, this calculation gives a single number that can be very telling about the matrix properties. A common formula to calculate the determinant for a 3x3 matrix is:\[det = a(ei - fh) - b(di - fg) + c(dh - eg)\]Here’s a breakdown of how to use this formula:

• Multiply and subtract the elements as per the pattern in the formula.
• Each element a, b, c serves as a scaling factor, modifying outcomes with their coefficient values.

In matrices where elements can only have values of 1 or -1, like in this exercise, calculating determinants involves plugging these numbers into the formula and checking which permutations give the minimum determinant result. Understanding determinants is vital because they illustrate key properties of matrices, such as invertibility and volume distortion in transformations. In the case of this exercise, finding the permutation that provides the smallest determinant is the problem's objective.