Problem 71 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\left|\begin{array}{ccc}x+3 & x+2 & (x+2)^{3} \\ x+2 & x+3 & (x+2)^{3} \\\ (x+2)^{3} & x+2 & x+3\end{array}\right|\) \(=a x^{7}+b x^{6}+c x^{5}+d x^{4}+e x^{3}+f x^{2}+g x+h\) be an iden- tity in \(x\), where \(a, b, c, d, e, f, g, h\) are independent of \(x\), then the value of \(g\) is (A) \(-213\) (B) 213 (C) 0 (D) None of these

Step-by-Step Solution

Answer

The value of \( g \) is 0.

Step 1: Simplify the Matrix

Consider the given matrix \( A = \begin{pmatrix} x+3 & x+2 & (x+2)^3 \ x+2 & x+3 & (x+2)^3 \ (x+2)^3 & x+2 & x+3 \end{pmatrix} \). Recognize that there is symmetry in the matrix and each row (and column) includes the element \( (x+2)^3 \). Start by simplifying the matrix with row operations to make it easier to calculate the determinant.

Step 2: Using Row Operations

Subtract the second row from the first row: \\( R_1 \to R_1 - R_2 \) to get: \\( \begin{pmatrix} 1 & -1 & 0 \ x+2 & x+3 & (x+2)^3 \ (x+2)^3 & x+2 & x+3 \end{pmatrix} \). Next, subtract the third row from the new first row: \\( R_1 \to R_1 - R_3 \), resulting in: \\( \begin{pmatrix} 1 & -1 & 0 \ x+2 & x+3 & (x+2)^3 \ 0 & 0 & 0 \end{pmatrix} \).

Step 3: Zero Determinant Observation

Notice that the third row is now all zeros. According to properties of determinants, if any row or column of a matrix is zero, then the determinant of the matrix is zero. Therefore, \( \det(A) = 0 \). Every coefficient must be zero, including the coefficient of \( x \), \( g \).

Step 4: Determine the Coefficient of \( x \)

Since the determinant is 0, the polynomial must equal zero for all coefficients. Thus, the coefficient of \( x \), \( g \), is 0. Compute further if needed to confirm that each polynomial term coefficient satisfies the identity.

Key Concepts

Matrix Simplification

Matrix simplification means breaking down complex matrices into simpler forms, making them easier to analyze or calculate. The initial step when you have a complex matrix with repeating patterns or elements, like the given matrix, is to identify any symmetries. These can often be exploited using mathematical operations to reduce the matrix's complexity.

In the exercise, the matrix contains elements that are similar or repetitive along rows and columns. This symmetry hints that you can apply basic operations, like adding or subtracting rows, to simplify it. Simplifying allows you to reveal hidden properties such as the determinant, which is easier to calculate once the matrix is in a more manageable form. Always look out for similar patterns in matrices to simplify your work.

Row Operations

Row operations are techniques used to transform a matrix into a more operable form without altering its essential properties. They involve adding, subtracting, or multiplying entire rows by numbers, which are allowable moves that can help simplify complex matrices.

In the problem, row operations are used to subtract the second row from the first. This step eliminates some of the complexities by zeroing out elements, making subsequent calculations simpler and leading to the observation in the next section. These operations are crucial to simplifying matrices especially when solving for determinants, inverse matrices, or systems of equations.

• Add one row to another
• Multiply a row by a constant
• Subtract one row from another

Understanding and applying these basic strategies can greatly enhance your problem-solving efficiency.

Zero Determinants

A determinant is a special number that can be calculated from a square matrix. It provides valuable insights into some properties of the matrix, like whether it's invertible. One critical observation in matrix theory is that if a matrix has a row or a column full of zeros, its determinant will be zero.

This is seen in the step where, after performing row operations, the matrix has an entire row of zeros. This quickly tells us that the determinant of the matrix is zero. This is a powerful aspect of determinants that allows us to immediately understand certain outcomes without further calculation. When the determinant is zero, it often implies the matrix is singular, meaning it does not have an inverse.

Polynomial Coefficients

Polynomial coefficients are the numbers or variables that multiply each term in a polynomial. In the context of matrices, especially when dealing with determinants that are expressed as polynomials, understanding these coefficients is essential to solving matrix-related problems.

For the given exercise, the matrix determinant is expressed in polynomial form. The task involves calculating the determinant and equating its expansion to zero, due to zero determinant property, to find these coefficients. Here, the coefficient of \( x \) is zero, implying that in the polynomial expression of the determinant, this term's contribution to the polynomial sum must be zero.

Understanding how these coefficients interact can help you solve for unknowns in many mathematical contexts, making polynomial coefficients a crucial concept when dealing with matrix determinants.