Question

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are positive and not all equal, then the value of determinant $$ \left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| \text { is ... } $$ (a) \(>0\) (b) \(\geq 0\) (c) \(<0\) \((\mathrm{d}) \leq 0\)

Short answer

The correct answer is (b) \(\geq 0\).

Step-by-step solution

Expand the determinant using cofactor expansion

We will expand along the first row using the cofactor expansion method. $$ \left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| = a\left|\begin{array}{ll} c & a \\ a & b \end{array}\right| - b\left|\begin{array}{ll} b & a \\ c & a \end{array}\right| + c\left|\begin{array}{ll} b & c \\ c & a \end{array}\right| $$

Calculate the value of each cofactor

Now, we calculate each cofactor: $$ a\left|\begin{array}{ll} c & a \\ a & b \end{array}\right| = a(bc - a^2) \\ b\left|\begin{array}{ll} b & a \\ c & a \end{array}\right| = b(ab - ac) \\ c\left|\begin{array}{ll} b & c \\ c & a \end{array}\right| = c(ac - bc) $$

Substitute the cofactors and simplify

Now, let's substitute the cofactors into the expanded determinant and simplify: $$ \left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| = a(bc - a^2) - b(ab - ac) + c(ac - bc) $$ Simplify the expression: $$ abc - a^3 - ab^2 + abc + ac^2 - bc^2 $$ Factor out common terms: $$ 2abc - a^3 + ac^2 - ab^2 - bc^2 $$

Compare the value to 0

Now we need to compare the expression to zero, so we need to decide if \(2abc - a^3 + ac^2 - ab^2 - bc^2 \geq 0\) or if \(2abc - a^3 + ac^2 - ab^2 - bc^2 < 0\). Since a, b, and c are all positive and not all equal, we can observe that: 1. \(a^3\) is always positive, as the product of three positive equal numbers is positive. 2. \(ab^2\), \(ac^2\), and \(bc^2\) are all positive as they are the product of positive numbers. 3. \(2abc\) is always positive since all values of a, b, and c are positive. Now, we must see if the expression as a whole can be negative or not. We can rewrite the expression as: $$ 2abc - a^3 + ac^2 - ab^2 - bc^2 = 2abc - (a^3 + ac^2 - 2abc) - (ab^2 - 2abc + bc^2) $$ From this, we can see that the two terms in parentheses represent incomplete squares, i.e., \((a - c)^2 - 4bc\) and \((a - b)^2 - 4ac\). Therefore, it's possible that they could be negative or zero. However, since a, b, and c are not all equal and all positive, one of these terms is guaranteed to be greater than zero. Therefore, the overall expression: $$ 2abc - a^3 + ac^2 - ab^2 - bc^2 \geq 0 $$ So, the value of determinant is greater than or equal to 0.

Answer

The correct answer is (b) \(\geq 0\).

Key concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a helpful method in matrix algebra used to calculate the determinant of a matrix. It involves breaking down a larger determinant into smaller ones that are easier to manage and compute.
This method uses minors and cofactors to achieve this simplification. Each minor is a smaller matrix formed by deleting a specific row and column, while cofactors give each term a sign depending on their position.
For the determinant of a 3x3 matrix like the one given in the exercise:

• Choose a row or column to expand along; the first row is often a convenient choice.
• Multiply each element in this row by the determinant of the 2x2 minor that remains after removing its row and column.
• Adjust the sign according to the cofactor sign pattern, alternating starting with positive.

In the provided solution, cofactor expansion is employed on the first row of the given determinant, leading to a series of smaller determinants that are easier to compute.

Matrix Algebra

Matrix algebra is a crucial branch of mathematics dealing with matrices, which are rectangular arrays of numbers or functions. In this context, matrix algebra enables computations such as solving systems of equations, performing transformations, and calculating determinants.
Understanding the properties and manipulations of matrices is vital for solving many mathematical problems. Some key concepts include:

• Matrix multiplication and addition, allowing the combination of matrices to form new matrices or modify existing ones.
• Inverse matrices, which reverse the effects of a transformation represented by a given matrix.
• The determinant, which is a scalar value revealing certain properties of a matrix, such as invertibility or volume scaling in transformations.

In our problem, the calculation of the matrix determinant through cofactor expansion is a straightforward application of matrix algebra.

Inequalities in Mathematics

Inequalities are an essential part of mathematics, providing a way to compare the size or value of different expressions. They play a critical role in determining what conditions hold true for certain variables, guiding us to possible solutions or constraints.
In mathematical inequalities, you might see symbols like:

• \( > \) "greater than"
• \( < \) "less than"
• \( \geq \) "greater than or equal to"
• \( \leq \) "less than or equal to"

For our specific exercise, the context is determining the sign (positive, negative, or zero) of the determinant value, represented as the inequality \( \geq 0 \). This involves analyzing the expression obtained from cofactor expansion and simplifying it. In doing so, one checks the possible states a determinant can be in, given the conditions that \(a\), \(b\), and \(c\) are positive and not all equal. The comparison helps us conclude the nature of the determinant, ensuring it satisfies the condition \( \geq 0 \).