Problem 49 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b \end{array}\right|=a^{3}+b^{3}+c^{3}-3 a b c $$

Step-by-Step Solution

Answer

To prove the given identity, expand the determinant using the rule of Sarrus, simplify the expression by expanding the products and combining like terms, and finally combine the terms to show that the value of the determinant is equal to \(a^3 + b^3 + c^3 - 3abc\).

Step 1: Expand the determinant

We will expand the determinant using the rule of Sarrus, which consists of taking the sum of the products of the elements of the the diagonals going from top-left to bottom-right, and subtracting the sum of the products of the elements of the diagonals going from top-right to bottom-left. For the given determinant, we have: $$ \begin{vmatrix} a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b \end{vmatrix} = (a(b-c)(a+b)) +(b(c-a)(c+a)) +(c(a-b)(b+c)) -(c(a-b)(b-c)) -(a(b+c)(c+a)) -(b(c+a)(a+b)) $$

Step 2: Simplify the expression obtained

Now, we will simplify this expression by expanding the products and combining like terms: $$ \begin{aligned} &(a(b-c)(a+b)) +(b(c-a)(c+a)) +(c(a-b)(b+c))\\ &-(c(a-b)(b-c)) -(a(b+c)(c+a)) -(b(c+a)(a+b))\\ &= a(b(a+b)-c(a+b)) + b(c(c+a)-a(c+a)) + c(a(b+c)-b(b+c))\\ &\quad - c(a(b-c)-b(b-c)) - a(b(c+a)-c(c+a)) - b(c(a+b)-a(a+b))\\ &= a^3 + ab^2 - ac^2 - abc + b^3 - ab^2 + bc^2 - abc + c^3 - ac^2 + bc^2 - abc - abc \end{aligned} $$

Step 3: Combine like terms and verify the identity

In this step, we will combine the like terms to simplify the expression further: $$ \begin{aligned} &= a^3 + b^3 + c^3 - 3abc\\ \end{aligned} $$ Now, we have shown that the value of the determinant is equal to the given identity \(a^3 + b^3 + c^3 - 3abc\). Therefore, the given identity has been proved.

Key Concepts

Matrix Algebra

Matrix Algebra is a fundamental concept in mathematics that deals with matrices and operations involving them. A matrix is essentially a rectangular array of numbers or expressions arranged in rows and columns. This concept is widely applied in various fields such as engineering, physics, and computer science for solving linear equations, among other things. Matrices can be added, subtracted, and multiplied, but one of the most intriguing operations is finding their determinants. The determinant of a square matrix is a special number that provides valuable information about the matrix, such as whether it has an inverse or the volume of a geometric object it describes. In the given exercise, we deal with a 3x3 matrix, and our task is to prove a certain identity using its determinant.

Sarrus Rule

The Sarrus Rule is a straightforward method to calculate the determinant of a 3x3 matrix. This rule is applicable only to matrices of this size and involves a specific approach for determining the determinant. To apply Sarrus Rule, we consider diagonals of the matrix.

• Start by replicating the first two columns of the matrix to the right of the determinant.
• Next, multiply the elements on the diagonals from the top-left to the bottom-right.
• Then, perform the same for the diagonals from the top-right to the bottom-left.
• Finally, subtract the sum of the products of the second set of diagonals from the first set."

In the exercise, Sarrus Rule is used for the determinant expansion, leading to a more extended algebraic expression which is then simplified step-by-step to reveal the identity we are trying to prove.

Determinant Properties

Determinants have several notable properties that make them useful for understanding matrices. First, one of the key features is that swapping two rows or columns of a matrix changes the sign of its determinant. Also, if all elements of a row or a column are zeros, the determinant is zero. Notably, a matrix with two proportional rows has a determinant of zero, indicating linear dependence.

• Another important property is the linearity of determinants in rows and columns, meaning they behave predictably under addition and scalar multiplication.
• Determinants are also invariant under transposition, keeping their value the same if rows and columns are switched.
• Finally, the determinant of a product of two matrices is the product of their determinants.

In our problem, using these properties helps in simplifying the expanded determinant expression. Specifically, combining like terms efficiently results in the desired identity \(a^3 + b^3 + c^3 - 3abc\), successfully proving it. Understanding these properties allows us to handle complex matrices with more ease.