Problem 59 - Algebra and Trigonometry

Question

Show that $$ \left|\begin{array}{lll}{1} & {x} & {x^{2}} \\ {1} & {y} & {y^{2}} \\ {1} & {z} & {z^{2}}\end{array}\right|=(x-y)(y-z)(z-x) $$

Step-by-Step Solution

Answer

The determinant simplifies to \((x-y)(y-z)(z-x)\) after proper factoring.

Step 1: Understand the problem

We are asked to find the determinant of a 3x3 matrix and show that it is equal to \((x-y)(y-z)(z-x)\). The matrix given is constructed using terms involving different variables in a polynomial sequence.

Step 2: Recall the formula for a 3x3 determinant

The determinant of a 3x3 matrix of the form \[\begin{vmatrix} a & b & c \d & e & f \g & h & i \end{vmatrix}\]is given by \(a(ei-fh) - b(di-fg) + c(dh-eg)\).

Step 3: Substitute the matrix elements into the determinant formula

Substituting the values from the matrix \[\begin{vmatrix} 1 & x & x^2 \1 & y & y^2 \1 & z & z^2 \end{vmatrix}\]into the determinant formula, we obtain:\[1(y\cdot z^2 - z\cdot y^2) - x(1\cdot z^2 - 1\cdot y^2) + x^2(1\cdot z - 1\cdot y)\]

Step 4: Simplify the expression

Perform the arithmetic on each component: - The first term gives \(y\cdot z^2 - z\cdot y^2\).- The second term is \(-x(z^2 - y^2)\), which can be written as \(-x(z-y)(z+y)\).- The third term is \(x^2(z-y)\).Combine all terms: \[y\cdot z^2 - z\cdot y^2 - x(z^2-y^2) + x^2(z-y)\].

Step 5: Factor the expression

Recognize common factors in the expression:- Notice that \(y\cdot z^2 - z\cdot y^2\) can be rewritten with factoring: \((y-z)(-yz)\).- Combine like terms to notice a pattern where grouping and factoring \((x-y)(y-z)(z-x)\) is possible.Verify that upon expanding any remaining groupings you get back to previous steps.

Step 6: Verify the final expression

Combine adjusted terms and verify the expression is identical to the given expression:\[(x-y)(y-z)(z-x)\].Each term simplifies to confirm that all original factors remain in product.

Key Concepts

Vandermonde Determinant

A Vandermonde matrix is a type of matrix where each row forms a geometric progression of the variables. A common example is a matrix structured as follows:

• First row: All entries typically start with 1, incrementing powers of a variable.
• Successive rows: Each row progresses with increasing powers of distinct variables.

For the given exercise, our Vandermonde matrix involves three variables: \(x\), \(y\), and \(z\), each raised to increasing powers across rows:\[\begin{pmatrix}1 & x & x^2 \1 & y & y^2 \1 & z & z^2 \\end{pmatrix}\]The fascinating aspect of determining a Vandermonde determinant lies in its ability to transform into a polynomial of differences between variable pairs. In this case, the determinant of the 3x3 matrix simplifies neatly to \((x-y)(y-z)(z-x)\).
This expression showcases polynomials' inherent symmetries and highlights the geometric relationships embodied by a Vandermonde matrix.

3x3 Matrix

Understanding matrices is essential in linear algebra. A 3x3 matrix consists of three rows and three columns, often denoted in a format like: \[\begin{pmatrix}a & b & c \d & e & f \g & h & i \\end{pmatrix}\]The determinant of a 3x3 matrix is a special scalar value, providing insights into various properties such as invertibility. The formula to determine this value involves a combination of products and differences:\[det = a(ei - fh) - b(di - fg) + c(dh - eg)\]Implementing this formula requires substituting specific entries from the matrix to evaluate its determinant. For our Vandermonde matrix:

• Calculate each product as per the formula's terms.
• Gather results to understand the determinant as a polynomial or simpler expression.

In the given problem, this process reveals hidden relationships between the variables' sequences. It simplifies into the expression \((x-y)(y-z)(z-x)\).

Polynomial Sequences

Polynomial sequences are formed when numbers are arranged according to a polynomial's degrees. In our matrix, each column follows a polynomial sequence:

• The first column represents the constant polynomial, showing a sequence of ones.
• The second column includes the first powers, forming a linear progression.
• The third column features squares, creating a quadratic series.

These sequences are key to understanding how polynomial relationships are manipulated within matrices.
Particularly, polynomial sequences help reveal transformation properties in matrices, like simplifying determinants or identifying factorable expressions. With variables \(x\), \(y\), and \(z\), you see polynomial sequences form structural paths to determine interactions at higher dimensions.
The exercise demonstrates that through polynomial sequences and their inherent symmetries, complex patterns can be unraveled to simpler, yet significant relationships such as \((x-y)(y-z)(z-x)\). This highlights the beauty and power of matrices and polynomials in capturing mathematical relationships in compact forms.