Problem 71 - Precalculus with Limits

Question

In Exercises 71-76, evaluate the determinant(s) to verify the equation. \(\left| \begin{array}{r} w & x \\ y & z \end{array} \right| = -\left| \begin{array}{r} y & z \\ w & x \end{array} \right|\)

Step-by-Step Solution

Answer

The equation is true as the det(left matrix) = -det(right matrix).

Step 1: Evaluate Left Side Determinant

Calculate the determinant of the left matrix: \( \text{Det}(w, x; y, z) = w*z - x*y \)

Step 2: Evaluate Right Side Determinant

Calculate the determinant of the right matrix: \( \text{Det}(y, z; w, x) = y*x - z*w \)

Step 3: Verify equation

The equation states that the first determinant is equal to the negative of the second one. So, if \( w*z - x*y = -(y*x - z*w) \), then the equation is true. Simplifying the right side of the equation we have \( w*z - x*y = -y*x + z*w \). As \( w*z = z*w \) and \( -x*y = -y*x \), they match, so the equation is proved to be true.

Key Concepts

Matrix

A matrix is a rectangular array of numbers arranged in rows and columns. It is a powerful tool in mathematics, particularly useful for solving systems of linear equations, performing linear transformations, and more. In a matrix, each element is identified by two numbers: the row number and the column number. For example, in a 2x2 matrix:

• The first number is the row.
• The second number is the column.

A simple way to represent a 2x2 matrix is as follows:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]This layout makes matrices easy to handle mathematically and visually. Matrices can vary in size, depending on the number of rows and columns they have. However, the essential concept remains the same.

Determinant of a Matrix

The determinant is a special number calculated from a square matrix. It plays a significant role in linear algebra, especially in solving systems of linear equations and finding the inverse of a matrix. The determinant provides a scalar value that can help determine if a matrix has a unique solution.For a 2x2 matrix:\[\begin{bmatrix}w & x \y & z\end{bmatrix}\]The determinant, usually denoted as \(|A|\), is calculated using the formula:\[w \cdot z - x \cdot y\]

• If the determinant is zero, the matrix does not have an inverse, and the system of equations may not have a unique solution.
• If the determinant is non-zero, the matrix is invertible, and there is a unique solution for the system.

The determinant concept is crucial, as it can also be extended to larger matrices, following more complex techniques.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a mathematical statement. It involves a sequence of statements, each logically following from the previous ones, to demonstrate a concept or formula. In the context of determinants, proofs often involve verifying algebraic properties or identities. In the exercise, the proof involves showing that the determinant of one matrix is equal to the negative determinant of another. To achieve this, we performed the following steps: - Calculated the determinant of each matrix separately. - Compared the results, verifying that the left side determinant equals the negative of the right side determinant. This process helps in understanding how different manipulations or rearrangements of a matrix affect its determinant. Mathematical proofs like this one are fundamental in establishing the properties and behaviors of algebraic objects like matrices.