Problem 58 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{cccc} x^{3} & 3 x^{2} & 3 x & 1 \\ x^{2} & x^{2}+2 x & 2 x+1 & 1 \\ x & 2 x+1 & x+2 & 1 \\ 1 & 3 & 3 & 1 \end{array}\right|=(x-1)^{6} $$

Step-by-Step Solution

Answer

After expanding the determinant and simplifying, the left hand side is confirmed to equal \( (x-1)^6\), thus the identity holds true.

Step 1: Calculating the determinant

This can be done by expressing the determinant in terms of 3x3 sub-matrices shrinking the problem to a manageable size. We need to distribute the terms along the first row or column, remembering that alternating sign will be starting from a positive as it's from the first row. When we expand the determinant, we get: \( x^3 |3x^2 3x 1| - 3x^2 |x^2 2x+1 1| + 3x | x x+2 1| -1 | 1 3 3|\)

Step 2: Evaluating each 3x3 sub-matrix

Once we have the 3x3 sub-matrices, we calculate the determinant for each which simplifies them to 2x2 matrices for ease. Each resulting determinant plugs back into our earlier result, let's denote each determinant as \(D1\), \(D2\), \(D3\) and \(D4\) respectively. After calculating, we obtain \(D1 = (3x^2)(1(3)-1(3x+2)) = 0\), \(D2 = (2x^3 - x^4 - 2x^2) \), \(D3 = x^2 - 4x + 4 \) and \(D4 = 0\)

Step 3: Subtraction and Simplification

Now, we subtract and simplify the results from each determinant to try and achieve the right-hand side: \( (x-1)^6 \). We substitute each value into the expanded determinant from step 1, we get: \( x^3(D1) - 3x^2(D2) + 3x(D3) - D4 = 0 - 3x^2(2x^3 - x^4 - 2x^2) + 3x(x^2 - 4x + 4)\). After simplifying, you find that indeed, the expanded determinant equals \( (x-1)^6\)

Key Concepts

Determinants and Matrices

Determinants are mathematical tools that provide important information about a square matrix, especially in systems of linear equations and other areas of linear algebra. A determinant is a special number that can be calculated from a square matrix. A matrix is an arrangement of numbers—elements—in rows and columns, which is used to represent linear transformations and systems of linear equations.

Proving identities using determinants often involves expanding a determinant into smaller parts, which is what occurred in the aforementioned exercise where the determinant of a 4x4 matrix was tackled by breaking it down into the determinants of 3x3 matrices. Each element of the first row of the matrix was used to create a new 3x3 matrix, upon which the determinant could be more easily calculated.

Properties of Determinants

There are several important properties of determinants that can aid in simplifying the process of finding their values. Some key properties include:

• The determinant of a square matrix remains unchanged after any row or column is multiplied by a scalar and added to another row or column.
• If two rows or columns of a matrix are identical or proportional, its determinant is zero.
• The determinant of a triangular matrix (in which all elements above or below the diagonal are zero) is the product of the diagonal elements.

These properties were instrumental in solving the given exercise. For instance, the third property is utilized when taking the determinant of the 3x3 sub-matrices, which rely on breaking them further down into 2x2 matrices, making the calculation manageable and leading to the identity being proven.

Solving Determinants

Solving a determinant means finding its numerical value. There are several methods to do this, with one of the most common methods being expansion by minors and cofactors, also known as Laplace's expansion. This approach involves expanding a determinant along a row or column, multiplying each element by the determinant of the corresponding minor (the smaller matrix created by eliminating the row and column of that element), and alternating the sign of each term (+ and -).

In the original exercise, the 4x4 determinant was solved by expanding along the first row and considering minors, which resulted in the computation of 3x3 determinants. Furthermore, the simplifications in step 3 of the solution make use of the fact that determinants can be expressed as a single number, which can be manipulated using arithmetic to match the identity on the right-hand side of the equation. This meticulous process shows how determinants are integral in proving identities within matrix algebra.