Problem 22 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 1 & b c & b c(b+c) \\ 1 & c a & c a(c+a) \\ 1 & a b & a b(a+b) \end{array}\right| $$

Step-by-Step Solution

Answer

The short answer for evaluating the determinant is: \[ \det\left(\begin{array}{lll} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{array}\right) = ca^2(c+a) - ab^2(a+b) - bcab(a+b-1) + bc^2(ca(c+a) - ca) \]

Step 1: Write down the matrix determinant formula for 3x3 matrices

For a 3x3 matrix, the determinant is calculated using the formula: \[ \det\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right) = a\left(ei - fh\right) - b\left(di - fg\right) + c\left(dh - eg\right) \] In our case, the elements of the matrix are given by: \[ a = 1,\ b = bc,\ c = bc(b+c),\\ d = 1,\ e = ca,\ f = ca(c+a),\\ g = 1,\ h = ab,\ i = ab(a+b) \]

Step 2: Replace the elements of the formula with the elements of our matrix

Now we need to plug in the values of our matrix into the determinant formula: \[ \det\left(\begin{array}{lll} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{array}\right) = 1\left(ca(ca(c+a)) - ab(ab(a+b))\right) - bc\left(1(ab(a+b)) - ab(1)\right) + bc(b+c)\left(1(ca(c+a)) - ca(1)\right) \]

Step 3: Simplify and compute the value of the determinant

Let's simplify and compute the value of the determinant by calculating each term individually: First term: \[ 1(ca(ca(c+a)) - ab(ab(a+b))) = ca^2(c+a) - ab^2(a+b) \] Second term: \[ -bc(1(ab(a+b)) - ab(1)) = -bc(ab(a+b) - ab) = -bcab(a+b-1) \] Third term: \[ bc(b+c)(1(ca(c+a)) - ca(1)) = bc(b+c)(ca(c+a) - ca) = bc^2(ca(c+a) - ca) \] Now we add these terms together to find the determinant: \[ \det\left(\begin{array}{lll} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{array}\right) = ca^2(c+a) - ab^2(a+b) - bcab(a+b-1) + bc^2(ca(c+a) - ca) \]

Key Concepts

Matrix Determinant Formula

Understanding how to work with matrices is fundamental in linear algebra, and evaluating determinants is a critical skill. The determinant of a matrix is a special number that can tell us things about the matrix, such as whether it has an inverse or what its volume distortion is. For a 2x2 matrix, the process is straightforward, but for a 3x3 matrix, we need to employ a specific formula.

The general matrix determinant formula for a 3x3 matrix looks like this:
\[\det\begin{pmatrix}a & b & c \ d & e & f \ g & h & i\end{pmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]
This formula involves multiplying and subtracting products of elements from the matrix, organized in a particular way. Always begin with the element in the top left position and alternate signs. Picture this process as covering up the row and column of each leading element, then calculating the determinant of the smaller 2x2 matrix that remains. This visually aids in grasping the pattern of expansion.

3x3 Matrix Determinant

Applying the determinant formula to a 3x3 matrix may seem daunting, but with practice, it becomes intuitive. When you're given a 3x3 matrix with variable expressions, like in our exercise, it's essential to substitute the elements into the formula carefully. Each element corresponds to a specific position in the determinant equation.

For instance, taking the elements of the given matrix and replacing them in our determinant formula results in a complex expression that requires attention to detail during simplification. The first step is translating the abstract elements of the matrix, like 'a', 'b', or 'c', into the actual value from the exercise. Following this, the computations involve addition, subtraction, and multiplication of these values.

To accurately solve for a variable determinant, every algebraic expression must be handled correctly. This requires familiarity with algebraic rules and the confidence to manipulate complex expressions safely.

Simplification of Algebraic Expressions

Simplification is a powerful tool in mathematics, turning intimidating equations into more manageable ones. To simplify algebraic expressions within a determinant, combine like terms and apply operations step by step, following the order of operations: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.

Using the expressions in our determinant calculation, we're not just dealing with numbers; we're also handling variables. This adds complexity, but the key is to systematically work through each part of the expression. Group like terms, distribute multiplication over addition, and factor expressions when possible to reduce them to their simplest form.

As seen in the exercise solution, breaking down the calculation into smaller, more digestible steps allows each part to be simplified before combining them to find the determinant. This meticulous approach prevents mistakes and ensures a clear path to the correct answer. Practical examples, such as the one provided, allow students to put theory into practice and build confidence in simplifying algebraic expressions in the context of determinants.