Problem 53 - Precalculus Mathematics for Calculus

Question

Solve for \(x\) $$\left|\begin{array}{lll} 1 & 0 & x \\ x^{2} & 1 & 0 \\ x & 0 & 1 \end{array}\right|=0$$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Calculate the Determinant

To solve for \(x\), first we need to calculate the determinant of the 3x3 matrix. Using the rule for 3x3 matrices, the determinant \(|A|\) of matrix \(A\) is given by: \[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\] Substituting the given elements of the matrix, we have: \[|A| = 1(1 \cdot 1 - 0 \cdot 0) - 0(x^2 \cdot 1 - 0 \cdot x) + x(x^2 \cdot 0 - x \cdot 1)\] This simplifies to: \[|A| = 1 \cdot 1 - x^3\] Therefore, the determinant is \(1 - x^3\).

Step 2: Set the Determinant Equal to Zero

Since the determinant must be zero, set the equation \(1 - x^3 = 0\). Now, solve this equation to find the possible values of \(x\).

Step 3: Solve for x

To solve \(1 - x^3 = 0\), add \(x^3\) to both sides: \[1 = x^3\] Take the cube root of both sides to solve for \(x\): \[x = 1\] Hence, \(x\) is equal to 1.

Key Concepts

3x3 Matrix

A 3x3 matrix is a rectangular array consisting of three rows and three columns. These matrices are a fundamental concept in linear algebra. It’s important because it can represent a system of three linear equations, provide transformations in a three-dimensional space, and is pivotal in computer graphics and engineering processes.
Each position in the matrix is filled with either a number, symbol, or variable, making it versatile in mathematical applications. The 3x3 matrix used in our exercise is:
\[ \begin{bmatrix} 1 & 0 & x \ x^{2} & 1 & 0 \ x & 0 & 1 \end{bmatrix} \]
In this matrix, observe that some elements are constants while others are variables such as \(x\) and \(x^2\). This arrangement is crucial in determining how the matrix behaves when you calculate the determinant or perform other matrix operations.

Solve for x

Solving for \(x\) in the context of a determinant involves a step-by-step process that begins with calculating the determinant as shown. The goal is to find values of \(x\) that will make the determinant equal to zero.
After calculating the determinant of the matrix as \(1 - x^3\), we set the expression equal to zero to solve for \(x\): \[ 1 - x^3 = 0 \]

• First, rearrange the equation to: \(1 = x^3\)
• Next, solve for \(x\) by taking the cube root on both sides: \(x = 1\)

By following these steps, we concluded that \(x = 1\) is the value that satisfies the equation. This method is not only essential in solving such determinant-related problems but also offers foundational steps for similar algebraic equations involving matrices.

Matrix Algebra

Matrix Algebra is a rich field in mathematics that deals with operations involving matrices, such as addition, subtraction, multiplication, and determining inverses and determinants. Understanding matrix algebra is crucial because it provides solutions to complex linear systems, transformations, and helps in various computations in engineering, physics, and computer science.
In our exercise, one important application is using matrix determinants to solve for variables like \(x\) when a determinant is set to zero. The process for calculating the determinant of a 3x3 matrix follows a specific pattern, employing the rule:\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]This rule helps simplify the matrix into a single scalar value, which indicates specific properties of the matrix, such as invertibility or singularity. Knowledge of these properties is key, as they are instrumental in solving systems of equations and modeling real-world phenomena.