Question

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

Short answer

\[(x-y)(y-z)(x-z)\]

Step-by-step solution

Identify the determinant

Consider the given determinant: ewline \[\text{A} = \begin{vmatrix} x^2 & x & 1 \ y^2 & y & 1 \ z^2 & z & 1 \ \ \end{vmatrix} \]

Perform row operations

Perform operations to simplify the determinant. Specifically, subtract the second row from the first and the third row from the second. This gives: ewline \[\begin{vmatrix} x^2-y^2 & x-y & 0 \ \ y^2-z^2 & y-z & 0 \ \ z^2 & z & 1 \ \end{vmatrix} = \begin{vmatrix} (x-y)(x+y) & x-y & 0 \ \ (y-z)(y+z) & y-z & 0 \ \ z^2 & z & 1 \ \end{vmatrix} \]

Factor out common terms

Factor out \(x-y\) from the first row, \(y-z\) from the second row, and \(z^2\) from the third row:ewline ewline \[ (x-y)(y-z)\begin{vmatrix} x+y & 1 & 0 \ y+z & 1 & 0 \ z & z & 1 \end{vmatrix} =(x-y)(y-z)\begin{vmatrix} x+y & 1 & 0 \ y+z & 1 & 0 \ z & z & 1 \end{vmatrix} \]

Simplify the matrix

The determinant of the simplified matrix can be further simplified since the column already has zeros except for the first two rows. Using the properties of determinants, omit these rows: ewline ewline \[(x-y)(y-z)\begin{vmatrix} z \ \ 1 \end{vmatrix} \]

Calculate the final determinant

Finally, expand the remaining determinant to reach the final expression. It simplifies and yields: ewline ewline \[ (x-y)(y-z)(x-z) \]

Key concepts

Row Operations

Understanding 'row operations' is crucial when working with determinants. Row operations involve manipulating the rows of a matrix to simplify computations or transform the matrix into a more manageable form. Common row operations include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a multiple of one row to/from another

These operations help in reducing the matrix to a simpler form, allowing for easier calculation of the determinant. In the given example, subtracting one row from another extracts common factors, simplifying the determinant calculation.

Matrix Simplification

Matrix simplification is the process of transforming a matrix into a more straightforward structure, often making it easier to compute determinants. Simplifying a matrix often involves using row operations to create zeros in specific positions. In our example, we perform operations to achieve zeros in the third column, which simplifies our matrix and makes it easier to factor out common terms. Simplification can reveal patterns and relationships within the matrix elements that are not immediately obvious in the original form.

Factorization

Factorization in the context of determinants involves breaking down a determinant into the product of simpler expressions. By factoring out common terms from rows and columns, we reduce the complexity of the determinant. For example, in our exercise, after performing row operations, we factor out terms like (x-y) and (y-z), simplifying the matrix further. This step is vital because it reduces the determinant to a form that is much easier to evaluate, especially when we're dealing with polynomials or larger matrices.

Properties of Determinants

Understanding the properties of determinants is crucial for simplifying and solving determinants efficiently. Some key properties include:

• If two rows (or columns) are identical, the determinant is zero
• Swapping two rows (or columns) changes the sign of the determinant
• Multiplying a row (or column) by a scalar multiplies the determinant by that scalar

These properties are applied in the given example to simplify the matrix and factor out terms. Recognizing these properties can greatly reduce the amount of computation needed and simplify complex determinant calculations.