Problem 3 - Precalculus

Question

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(Q=\left[\begin{array}{ll}x & x^{2} \\ 1 & 2 x\end{array}\right]\)

Step-by-Step Solution

Answer

The determinant of matrix \( Q \) is \(-x^2 \)."

Step 1: Identify the Formula

To find the determinant of a 2x2 matrix \( Q \), we use the formula: \( \text{det}(Q) = ad - bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix positioned as follows: \[ Q = \begin{bmatrix} a & b \ c & d \end{bmatrix} \].

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The determinant of matrix \( Q \) is \(-x^2 \)."

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices in linear algebra. It consists of two rows and two columns, creating a total of four elements. Each element is identified by its row and column position. For example, in the matrix \( Q = \begin{bmatrix} x & x^2 \ 1 & 2x \end{bmatrix} \), \( x \) is located in the first row and first column, \( x^2 \) in the first row and second column, 1 in the second row and first column, and \( 2x \) in the second row and second column. This layout is crucial for performing specific matrix operations, such as calculating the determinant. The arrangement in a 2x2 matrix looks like this:

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

Understanding this structure helps in correctly applying formulas and performing matrix calculations.

Basics of Linear Algebra

Linear algebra is a field of mathematics that focuses on vectors, matrices, and linear transformations. It's essential for solving systems of linear equations and is widely used in various scientific disciplines, including computer science, engineering, and physics. In linear algebra, matrices are used to represent and solve equations. A matrix can be thought of as a collection of numbers arranged in rows and columns which can represent multiple linear equations. For instance, in the equation \( Ax = b \), \( A \) is a matrix, \( x \) is a vector of variables, and \( b \) is a vector of solutions. Matrices can interact with vector spaces in many ways, such as by scaling vectors or rotating space, and they form the basis for numerous complex operations in linear algebra, making them fundamental to understanding the subject.

Determinant Formula for 2x2 Matrices

The determinant of a 2x2 matrix is an important value that provides information about the matrix. The determinant can indicate if the matrix is invertible, and it also reveals important properties such as scaling in transformations. For a 2x2 matrix \( Q = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is found using the formula: \[\text{det}(Q) = ad - bc \] This simple formula involves multiplying the elements of the diagonal (\( a \) and \( d \)) and subtracting the product of the off-diagonal elements (\( b \) and \( c \)). This calculation is relatively straightforward, but it has profound implications in determining the behavior of matrices. When the determinant is zero, the matrix does not have an inverse, indicating that it cannot fully transform vector spaces and that some information might be lost in calculations. Conversely, a non-zero determinant means that the matrix is invertible, preserving the full structure of vector manipulations.