Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve for \(x\) $$\left|\begin{array}{ll} x & 2 \\ 1 & x \end{array}\right|=2$$

Step-by-Step Solution

Answer

The solutions for \(x\) are 2 and -2.

Step 1: Write Out the Determinant Equation

The determinant of a 2×2 matrix \(\left|\begin{array}{ll} a & b \ c & d \end{array}\right|\) is defined as \(ad - bc\). Thus, the given equation can be rewritten as \(x* x - 2* 1 = 2\) or simply \(x^2 - 2 = 2\).

Step 2: Solve the Equation for x

Solve the equation from the previous step for \(x\). Here it simplifies to \(x^2 -2 = 2\). Bring '2' to the other side to make it equal to zero, the equation then becomes \(x^2 - 4 = 0\). Then solve for \(x\). This gives us the solutions \(x = 2\) and \(x = -2\).

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It is a foundational subject in mathematics because it allows us to describe and solve problems in geometry, physics, economics, and many other fields using a common language of linear equations and matrices.

For instance, the exercise given involves a system represented by a single quadratic equation obtained from a determinant, showcasing how linear algebra can extend beyond linear systems to encompass quadratic relationships as well.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It provides important information about the matrix, like whether a set of linear equations has a unique solution or not. In the case of a 2x2 matrix, the determinant is found using the formula: \( ad - bc \).

For the exercise given, the matrix's determinant gives rise to a quadratic equation. Here, the process of finding the determinant leads to understanding the properties of the matrix and is crucial in solving for the variable \(x\). Understanding how to compute a determinant is a key component of many operations in linear algebra, including finding the inverse of a matrix or testing a matrix's singularity.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \). These equations describe parabolas when graphed and are foundational in varied areas of mathematics and science.

When solving determinant equations that result in quadratic form, as in the exercise, we apply the same techniques used for solving any quadratic equation: finding factors, completing the square, or using the quadratic formula. In the problem provided, factoring is used to find the values of \(x\) that satisfy the equation \(x^2 - 4 = 0\), leading to the solutions \(x = 2\) and \(x = -2\).