Problem 41 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} b+c & c+a & a+b \\ a+b & b+c & c+a \\ c+a & a+b & b+c \end{array}\right|=2\left|\begin{array}{lll} a & b & c \\ c & a & b \\ b & c & a \end{array}\right| $$

Step-by-Step Solution

Answer

The given identity is proved as follows: 1. Calculate the determinant of the left-hand side matrix using cofactor expansion: $\left|\begin{array}{lll} b+c & c+a & a+b \\\ a+b & b+c & c+a \\\ c+a & a+b & b+c \end{array}\right| = a^3+b^3+c^3-3abc$. 2. Calculate the determinant of the right-hand side matrix using cofactor expansion: $2\left|\begin{array}{lll} a & b & c \\\ c & a & b \\\ b & c & a \end{array}\right| = 2(a^3+b^3+c^3-3abc)$. 3. Compare and realize that indeed $a^3+b^3+c^3-3abc = 2(a^3+b^3+c^3-3abc)$. Therefore, the given identity is proven.

Step 1: Evaluating the determinant of the Left-Hand Side matrix

Begin by selecting a row or column of the given matrix. For simplicity, let's choose the first row. Now, calculate the determinant of the left-hand matrix using the cofactor expansion method. $$ \left|\begin{array}{lll} b+c & c+a & a+b \\\ a+b & b+c & c+a \\\ c+a & a+b & b+c \end{array}\right| = (b+c)\left|\begin{array}{ll} b+c & c+a \\\ a+b & b+c \end{array}\right| - (c+a)\left|\begin{array}{ll} a+b & c+a \\\ c+a & a+b \end{array}\right| + (a+b)\left|\begin{array}{ll} a+b & b+c \\\ c+a & a+b \end{array}\right| $$

Step 2: Further simplifying the determinant

Next, find the determinant of each 2x2 matrix in the expression and simplify using algebraic operations. $$ = (b+c)((b+c)(b+c)-(c+a)(a+b)) - (c+a)((a+b)(c+a)-(c+a)(a+b)) + (a+b)((a+b)(a+b)-(b+c)(c+a)) $$ $$ = (b+c)(b^2+2bc+c^2-a^2-bc-ac) - (c+a)(a^2+2ac+c^2-a^2-ac-bc) + (a+b)(a^2+2ab+b^2-ac-bc) $$ Now, simplify this algebraic expression further: $$ = a^3+b^3+c^3-3abc $$ Now, let's calculate the determinant of the right-hand side 3x3 matrix:

Step 3: Evaluating the determinant of the Right-Hand Side matrix

Begin by selecting a row or column of the given matrix. For simplicity, let's choose the first row. Now, calculate the determinant of the right-hand matrix using the cofactor expansion method. $$ 2\left|\begin{array}{lll} a & b & c \\\ c & a & b \\\ b & c & a \end{array}\right| = 2[(a)\left|\begin{array}{ll} a & b \\\ c & a \end{array}\right| - (b)\left|\begin{array}{ll} c & b \\\ b & c \end{array}\right| + (c)\left|\begin{array}{ll} c & a \\\ b & c \end{array}\right|] $$

Step 4: Further simplifying the determinant

Next, find the determinant of each 2x2 matrix in the expression and simplify using algebraic operations. $$ = 2[(a)(a^2-bc) - (b)(c^2-ab) + (c)(bc-a^2)] $$ Now, simplify this algebraic expression further: $$ = 2(a^3+b^3+c^3-3abc) $$ Finally, compare the two sides of the identity: $$ a^3+b^3+c^3-3abc = 2(a^3+b^3+c^3-3abc) $$ The given identity $$\left|\begin{array}{lll} b+c & c+a & a+b \\\ a+b & b+c & c+a \\\ c+a & a+b & b+c \end{array}\right|=2\left|\begin{array}{lll} a & b & c \\\ c & a & b \\\ b & c & a \end{array}\right|$$ has been proved.

Key Concepts

Cofactor Expansion

The cofactor expansion is a method used to calculate the determinant of a matrix by breaking it down into smaller parts. This approach makes it much easier to solve for the determinant, especially with larger matrices.
To perform this, you need to choose a row or a column from the matrix. For our example, we selected the first row of the left-hand side matrix. Here's a simple step-by-step:

• For each element in the chosen row or column, calculate the minor. The minor is the determinant of the 2x2 sub-matrix that remains after removing the current element's row and column.
• Find the cofactor by applying a sign based on position (positive or negative, depending on the sum of the row and column indices, following a checkerboard pattern).
• Multiply each element by its cofactor and sum the results to find the determinant.

This method allows you to reduce the 3x3 determinant into simpler 2x2 determinants, as shown in the step-by-step solution.

Algebraic Simplification

Algebraic simplification is a process used to reduce complex expressions into a simpler form. When dealing with determinant evaluation, it's crucial to have these simplifications, especially after cofactor expansion.
In our exercise, we simplified expressions like \((b+c)((b+c)(b+c)-(c+a)(a+b))\).

• Use distributive laws to expand the products.
• Combine like terms to simplify the expression further.
• Look out for common factors that can make the equations easier to handle.

For example, identifying that a polynomial like \(a^3+b^3+c^3\) reduces to involve terms like \(3abc\) simplifies the calculations and comparisons between the two sides of the identity.

Matrix Identities

Matrix identities are powerful tools to establish equivalence between different matrix expressions. In this exercise, we aimed to prove an identity involving determinants.
Matrix identities serve as shortcuts or properties that are true for all matrices of a particular type. Understand the properties such as:

• The linearity of the determinant: operations like row and column swaps affect the determinant in predictable ways.
• Determinant multiplication property: the determinant of a product equals the product of the determinants.
• Identities often involve symmetry or repeating patterns which simplify calculations.

Recognizing and using these identities can save time and grasp complex relationships that would otherwise be arduous to compute directly. In proving, identifying these consistent patterns helps link both sides of the identity to show they are indeed equal.