Problem 2 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 0 & -1 \\ 2 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 2.

Step 1: Identify the Matrix Type

The given matrix is a 2x2 matrix, represented as \(\begin{bmatrix}0 & -1 \ 2 & 0\end{bmatrix}\). For a 2x2 matrix, we use the specific formula for finding the determinant.

Step 2: Apply the Formula for a 2x2 Matrix

The determinant of a 2x2 matrix \(\begin{bmatrix}a & b \ c & d\end{bmatrix}\) is calculated using the formula \(ad - bc\). In this case, \(a = 0\), \(b = -1\), \(c = 2\), and \(d = 0\).

Step 3: Substitute Values into the Formula

Substitute the values into the determinant formula: \(0 \cdot 0 - (-1) \cdot 2\).

Step 4: Simplify the Expression

Calculate the expression: \(0 \times 0 = 0\) and \((-1) \times 2 = -2\). The formula simplifies to \(0 - (-2)\).

Step 5: Final Calculation

Simplify the result of the previous step: \(0 - (-2) = 0 + 2 = 2\). Thus, the determinant of the matrix is 2.

Key Concepts

2x2 Matrix

A 2x2 matrix, as the name suggests, is a matrix with two rows and two columns. It is usually represented in the form of:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Where:

• "a" and "b" are the elements of the first row.
• "c" and "d" make up the second row.

All matrices are essentially an array of numbers arranged in rows and columns. The purpose of matrices in mathematics, particularly in linear algebra, is vast:

• They are used to represent linear transformations, like rotating or scaling objects.
• They assist in solving systems of linear equations.
• They serve as a foundation for vector spaces, a crucial concept in linear algebra.

Understanding how to manipulate matrices like our 2x2 example, is key to unlocking more complex mathematical concepts and their applications.

Matrix Determinant Formula

The matrix determinant is a special number that can be calculated from a square matrix. For our example of a 2x2 matrix, the determinant is found using the formula:\[D = ad - bc\]Where:

• "a" and "d" are the diagonal elements of the matrix.
• "b" and "c" are the off-diagonal elements.

By plugging each number into the formula with our matrix:\[\begin{bmatrix} 0 & -1 \ 2 & 0 \end{bmatrix}\]we identify:

• "a" = 0
• "b" = -1
• "c" = 2
• "d" = 0

Plugging into the formula results in:\[0 \cdot 0 - (-1) \cdot 2 = 0 + 2 = 2\]The determinant of 2 is a value that provides us with several insights into the matrix's properties. For one, it tells us if the matrix is invertible (which it is, since the determinant is non-zero). This makes determinants a useful tool for solving matrix equations and understanding geometry transformations.

Linear Algebra Concepts

Linear algebra is a fundamental area of mathematics focused on vectors, vector spaces, linear transformations, and systems of linear equations. Matrices, like the 2x2 example, are central to these studies. Here are some important aspects:

• Linear transformations: Matrices represent these transformations, allowing us to perform operations like rotation and reflection in vector spaces.
• Matrix operations: Including addition, subtraction, and multiplication, are essential skills to analyze and simplify equations.
• Determinants: Not just limited to 2x2 matrices, determinants have a broad applicability in higher dimensions and complex calculations.

In real-world applications, linear algebra is used in various fields:

• Physics uses it to explain phenomena like wave dynamics, electricity, and optics.
• Computer science leverages it in algorithms, graphics, and machine learning.
• Economics uses it for modeling systems and predicting outcomes.

Thus, grasping basic linear algebra concepts equips students with tools applicable across technology, science, and engineering domains.