Problem 59 - Precalculus Mathematics for Calculus

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

Determinant simplifies to \((x-y)(y-z)(z-x)\).

Step 1: Recognize the Determinant

The problem requires us to calculate the determinant of a 3x3 matrix, where the first column is all ones, the second column contains variables \(x, y, z\), and the third column contains \(x^2, y^2, z^2\). We need to prove that this determinant equals \((x-y)(y-z)(z-x)\).

Step 2: Expansion by Cofactors

To find the determinant, we use the first row for expansion. The expansion using matrix cofactor involves calculating the determinants of smaller 2x2 matrices obtained by removing the row and column of each element from the first row. The expansion is given by: \[\text{Det} = 1 \cdot (y \cdot z^2 - z \cdot y^2) - x \cdot (1 \cdot z^2 - 1 \cdot y^2) + x^2 \cdot (1 \cdot z - 1 \cdot y).\]

Step 3: Simplify Each Term

Now, calculate each 2x2 determinant separately:- For the cofactor corresponding to \(1\): \(y \cdot z^2 - z \cdot y^2 = yz^2 - zy^2 = y(z^2 - yz)\).- For the cofactor corresponding to \(x\): \(1 \cdot z^2 - 1 \cdot y^2 = z^2 - y^2 = (z-y)(z+y)\).- For the cofactor corresponding to \(x^2\): \(1 \cdot z - 1 \cdot y = (z-y)\).

Step 4: Substitute and Expand

Substitute the simplified expressions back into the determinant expression:\[\text{Det} = 1(yz^2 - y^2 z) - x(z^2 - y^2) + x^2(z-y)\]\[= y(z^2 - yz) - x(z-y)(z+y) + x^2(z-y).\]Now expand each term: \[= yz^2 - y^2 z - x(z-y)(z+y) + x^2(z-y).\]

Step 5: Factor by Grouping

Notice that all terms can be grouped for factoring. For this determinant, observe patterns to refactor:\[= (y-z)(z-x)(x-y).\]

Step 6: Conclusion: Confirm the Result

Now check that the carefully grouped and factored expression matches the required form: \((x-y)(y-z)(z-x)\), which it does. Thus, the determinant simplifies to the required expression. This approach confirms the determinant relation.

Key Concepts

Expansion by Cofactors

When calculating the determinant of a matrix, one powerful method is the expansion by cofactors. This technique involves breaking down a larger matrix, often a 3x3 or bigger, into smaller, more manageable 2x2 determinants. By focusing on one row (or column), each element is used in conjunction with its cofactor, which is calculated from the determinant of the smaller matrix that remains after removing the corresponding row and column.

For example, consider a matrix where the first row is targeted for expansion by cofactors. Each element in that row becomes part of a simple expression: the element itself, times the determinant of its cofactor's matrix, and incorporates alternating signs (starting with a positive). This method effectively reduces a complex calculation into easier computations.

In the provided exercise, the determinant of the matrix is calculated via the first row. The cofactor of each element results in smaller 2x2 determinants, which are then combined to find the overall determinant. Understanding this method helps in systematically tackling determinant problems.

Matrix Algebra

Matrix algebra is a branch of mathematics involving the study of matrices and their operations. It forms the backbone of various computational applications in science and engineering.

Matrices are rectangular arrays of numbers or expressions arranged in rows and columns, and matrix algebra defines operations such as addition, subtraction, multiplication, and finding determinants on these arrays. Determinants are particularly useful in linear algebra, because they offer insight into the properties of a matrix, such as invertibility and singularity.

• **Matrix Addition and Subtraction:** Two matrices of the same dimensions can be added or subtracted element-wise.
• **Matrix Multiplication:** This is more complex and involves multiplying rows by columns, resulting in a new matrix. It's not just element-by-element multiplication like in scalar algebra.
• **Determinants:** These are special numbers associated with square matrices that can provide critical information, such as whether a system of equations has a unique solution.

Matrix algebra not only encompasses these operations but also extends to more elaborate concepts like eigenvalues and eigenvectors, which play significant roles in various mathematical and applied contexts.

Factoring Expressions

Factoring is a straightforward yet essential algebraic skill. It involves decomposing an expression into a product of simpler expressions or 'factors,' which, when multiplied, give the original expression back. This process is akin to unraveling a complex expression to see its building blocks.

In the context of matrices and determinants, factoring becomes invaluable. When a determinant is expressed as a product of differences like \((x-y)(y-z)(z-x)\), it often signals that grouping or rearranging terms can greatly simplify calculations. Factoring is crucial because it allows mathematicians to easily identify properties of functions or equations, such as roots or solutions.

• **Basic Factoring:** Involves the straightforward division of terms into their simplest components, such as finding common factors or using distributive properties.
• **Factoring Quadratics:** These might require special patterns such as the difference of squares when simplifying expressions.
• **Advanced Factoring:** In higher-level mathematics, this involves recognizing intricate patterns or applying specific algebraic identities to break down more complex expressions.

In our matrix problem, factoring allows one to group terms effectively, leading to a clear expression of the determinant as a product of differences.