Problem 1 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 6.

Step 1: Identify the Matrix Type

The matrix given is a 2x2 matrix: \[ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \]. A determinant exists for any square matrix.

Step 2: Use the Formula for 2x2 Matrices

The determinant \( \text{det}(A) \) of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula \( ad - bc \).

Step 3: Plug in the Values

Substitute the values from the matrix into the formula: \( a = 2 \), \( b = 0 \), \( c = 0 \), \( d = 3 \).

Step 4: Calculate the Determinant

Using the formula, compute: \( 2 \times 3 - 0 \times 0 = 6 \).

Step 5: State the Determinant

The determinant of the given matrix is 6.

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple array of numbers arranged in two rows and two columns. In mathematics, particularly in linear algebra, matrices such as this are fundamental tools for solving problems regarding systems of linear equations, transformations, and more. A 2x2 matrix follows this generic structure:

• The first row typically contains values: a and b
• The second row contains values: c and d

In written form, it appears as: \[\begin{bmatrix}a & b\c & d\end{bmatrix}\]This matrix may seem simple compared to larger matrices, but understanding it is crucial for grasping more complex linear systems. From a 2x2 matrix, one can also derive critical properties like the determinant, which is essential in matrix algebra.
Identifying matrix type is the first step in solving any matrix-based problem because different matrices come with specific methods and formulas for solving them.

Matrix Algebra

Matrix algebra involves various operations and properties specific to matrices. These operations include addition, subtraction, multiplication, and finding determinants. Each plays a significant role in different applications. Here's a deeper look:

• Matrix Addition and Subtraction: Can only be performed on matrices of the same dimensions. It’s a simple operation where you add or subtract corresponding elements.
• Matrix Multiplication: Involves rows and columns where the elements of the rows of the first matrix are multiplied with the corresponding elements of the columns of the second, then added together.
• Determinant of a Matrix: A scalar that provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix \[\begin{bmatrix} a & b\ c & d \end{bmatrix}\], it is calculated as \(ad-bc\).

Understanding these operations is vital, especially when dealing with systems of linear equations and transformations. Matrix algebra serves as the backbone for numerous scientific and engineering calculations.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It encompasses everything regarding matrices, including the concepts of determinants, eigenvalues, and eigenvectors.
Some fundamental aspects of linear algebra include:

• Vector Spaces: These are sets where vectors can be added together and multiplied by scalars to produce another vector in the same set.
• Linear Transformations: Functions that map vectors to other vectors in a linear manner, represented by matrices.
• Determinants: A key concept in linear transformations and matrix algebra, determinants can indicate if a matrix can be inverted and more.
• Systems of Linear Equations: Often solved using matrices and determinants, showcasing the intersection of various branches of linear algebra.

Grasping linear algebra's core concepts is essential for understanding how to solve equations, work with matrices, and apply these skills in various fields, from computer science to physics.