Problem 77 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\Delta(x)=\left|\begin{array}{ccc}x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6\end{array}\right|=A x^{4}+B x^{3}+C x^{2}+D x+E\)

Step-by-Step Solution

Answer

\(A = 1\), \(B = -1\), \(C = -12\), \(D = 12\), \(E = 0\).

Step 1: Understanding the Determinant Formula

Recall that the determinant of a 3x3 matrix \(A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\) is computed using the formula: \(\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\). We will apply this formula to the given matrix.

Step 2: Substitute and Apply Determinant Formula

The given matrix is \(\begin{pmatrix} x & 2 & x \ x^2 & x & 6 \ x & x & 6 \end{pmatrix}\). Start by computing: 1. \(ei - fh = x \cdot 6 - 6 \cdot x = 0\),2. \(di - fg = x^2 \cdot 6 - 6 \cdot x = 6x^2 - 6x\),3. \(dh - eg = x^2 \cdot x - x \cdot x = x^3 - x^2\).Substitute these into the determinant formula to simplify.

Step 3: Calculate Determinant Components

Now substitute each calculated term:\(x \cdot 0 - 2 \cdot (6x^2 - 6x) + x \cdot (x^3 - x^2)\),which simplifies to:\(-12x^2 + 12x + x^4 - x^3 \).

Step 4: Arrange Terms by Degree

Rearrange the terms in decreasing order of degree:\(x^4 - x^3 - 12x^2 + 12x\).

Step 5: Identify Coefficients

From the expression \(x^4 - x^3 - 12x^2 + 12x\), identify\(A = 1\), \(B = -1\), \(C = -12\), \(D = 12\), and \(E = 0\) (since there is no constant term).

Key Concepts

Matrix Algebra

Matrix algebra is an extension of regular algebra to matrices, which are rows and columns of numbers like a rectangular grid. Understanding how matrices interact with each other is key to solving many mathematical problems in various fields, including engineering and computer science.

Key operations in matrix algebra include addition, subtraction, multiplication, and finding the determinant. The determinant is a scalar value derived from a square matrix that provides useful information about the matrix such as whether it is invertible or not.

To calculate the determinant of a 3x3 matrix, you use a specific formula that involves the elements of the matrix. This formula allows you to convert a matrix into a single number. Here's the determinant formula for a 3x3 matrix:

• For a matrix \[ \begin{pmatrix}a & b & c \d & e & f \g & h & i \end{pmatrix} \] The determinant is calculated as: \[ det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

By following this formula, you can determine a polynomial expression that represents the determinant of a matrix. Each row and column of a matrix contributes to this polynomial, hence providing insights into the matrix’s properties.

Polynomial Expression

Polynomial expressions are mathematical phrases involving a sum of powers in one or more variables multiplied by coefficients. Commonly, they are written in the form of\[ax^n + bx^{n-1} + \, ... \, + zx^0\]where each letter represents a constant coefficient and \(x\) represents a variable. In the original exercise, our task is to express the determinant of a matrix as a polynomial in terms of \(x\).

When calculating the determinant, we discovered that it simplifies to the polynomial expression \(x^4 - x^3 - 12x^2 + 12x\). Now, we can identify the coefficients of the polynomial:\(A = 1\), \(B = -1\), \(C = -12\), \(D = 12\), and \(E = 0\). These coefficients represent the scalar factors of each respective term in the polynomial.

Understanding polynomials is crucial as they are everywhere in mathematics. Whether it's solving quadratic equations or modeling real-world scenarios via curves, polynomials play a vital role in technology and science to predict fully and explain natural phenomena.

Matrix Operations

Matrix operations refer to the various calculations you can perform involving matrices, such as addition, subtraction, and multiplication. These are fundamental processes that extend regular arithmetic operations to matrices. However, there are some important rules that differentiate them from scalar arithmetic, which you must adhere to.

When we talk about operations like determining a matrix's determinant, we are looking at how these values provide information about the matrix itself. Here are some important matrix operations you might encounter:

• Addition and Subtraction: You can add or subtract matrices of the same size by adding or subtracting corresponding elements.
• Multiplication: The rows of the first matrix are multiplied with the columns of the second matrix. The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to occur.
• Determination: Using a specific formula to produce a single number, providing insight into matrix properties.
• Inversion: Finding a matrix such that when it's multiplied by the original matrix, it results in the identity matrix.

Mastering these operations is crucial for handling complex matrix algebra tasks. Whether you're manipulating data in computer graphics or solving systems of equations in engineering, understanding these operations is essential.