Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rl}-0.5 & 2 \\\x & x\end{array}\right]=0$$

Step-by-Step Solution

Answer

x = 0.

Step 1: Set Up the Determinant Equation

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated as \(ad - bc\). For the given matrix \(\begin{bmatrix} -0.5 & 2 \ x & x \end{bmatrix}\), set up the determinant equation: \(-0.5 \cdot x - (2 \cdot x) = 0\).

Step 2: Simplify the Determinant Equation

Simplify the determinant equation from Step 1: \(-0.5x - 2x = 0\).

Step 3: Combine Like Terms

Combine the terms: \(-0.5x - 2x\) to get \(-2.5x\). This simplifies the equation to \(-2.5x = 0\).

Step 4: Solve for x

To solve \(-2.5x = 0\), divide both sides of the equation by -2.5: \(x = 0\).

Step 5: Verify the Solution

Substitute \(x = 0\) back into the original determinant expression to verify: \((-0.5 \cdot 0) - (2 \cdot 0) = 0\). The determinant is zero, confirming that \(x = 0\) is correct.

Key Concepts

Understanding the 2x2 Matrix

In the world of matrices, the 2x2 matrix is among the simplest yet important structures. A 2x2 matrix is a rectangular array of numbers arranged into 2 rows and 2 columns. Here's a generic form to visualize:

• First row: \(a, b\)
• Second row: \(c, d\)

Such a matrix can be represented as:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]This fundamental setup helps in many linear algebra calculations, including those involving determinants and solving linear equations. Remember that each element (a, b, c, and d) is just a value that can vary based on the specific problem being addressed.In solving equations involving determinants, the arrangement of these elements is crucial. It determines how you calculate the determinant and, ultimately, solutions for variables like \(x\) embedded in the matrix.

Working with Matrix Equations

Matrix equations involve calculations with matrices to find unknowns. They are equations in which matrices play a critical role, either through their arrangement or involvement in determinant calculations.Consider this example using a 2x2 matrix:\[\operatorname{det}\left[\begin{array}{cc}-0.5 & 2 \x & x\end{array}\right] = 0\]The goal is to solve this matrix equation to find \(x\). To do so, we look at the determinant, which is a special number calculated from a matrix.The determinant for a 2x2 matrix is determined by the formula:\[ad - bc\]where \((a, b, c, d)\) are the elements of the matrix. By plugging these values from our example, the equation becomes:\(-0.5 \cdot x - 2 \cdot x = 0\). This simplifies to help us find the value of \(x\). The matrix equation approach can efficiently solve problems that involve finding unknowns when matrices and their arrangements are involved.

Solving Determinant Equations

Solving equations that involve determinants, especially for a 2x2 matrix, follows a systematic process. The determinant value provides a pathway to solve for unknowns within the matrix elements.Let's break down the steps:

• Firstly, express the determinant equation. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), it's represented as \(ad - bc = 0\) if the determinant equals zero.
• Then, substitute the variables into this expression. In the given example, \(-0.5 \cdot x - 2 \cdot x = 0\)\ occurs.
• Combine like terms to simplify the expression: here, \(-2.5x = 0\).
• Finally, solve for the variable: divide both sides by the coefficient to isolate the unknown, giving \(x = 0\).

To verify the result, substitute back into the original expression to ensure the determinant holds true, confirming the solution for \(x\). These steps depict how determinant calculations directly lead to solutions in matrix equations, highlighting their crucial role in matrix algebra.