Problem 10 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} -2 & 1 \\ 3 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 1.

Step 1: Recall the Determinant Formula for a 2x2 Matrix

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \text{det}(A) \) is calculated as \( ad - bc \).

Step 2: Identify Elements of the Matrix

The given matrix \( \begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix} \) has elements \( a = -2 \), \( b = 1 \), \( c = 3 \), and \( d = -2 \).

Step 3: Apply the Formula

Substitute the identified elements into the determinant formula: \( (-2)(-2) - (1)(3) \).

Step 4: Calculate the Determinant

Perform the multiplication and subtraction: \( 4 - 3 = 1 \). The determinant of the matrix is 1.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves calculating the product of two matrices by following specific rules. This process is crucial when working with matrices because it allows you to combine them to form new matrices. Unlike regular multiplication of numbers, matrix multiplication is not straightforward and requires the following steps:

• Consider two matrices: the first matrix with dimensions of \( m \times n \) and the second matrix with dimensions of \( n \times p \).
• Ensure that the number of columns in the first matrix equals the number of rows in the second matrix, otherwise, the multiplication is not possible.
• Multiply each element of the rows of the first matrix by each element of the columns of the second matrix.
• Add the results of these multiplications to get the elements of the resulting matrix, which will have dimensions \( m \times p \).

Matrix multiplication is used extensively in computer graphics, physics simulations, and solving systems of linear equations. It offers a way to alter the dimensions or transform matrices into a different basis, and it's a crucial technique for linear transformations.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, matrices, and linear transformations. The study of linear algebra involves understanding lines and planes and how they intersect and interact. It forms the backbone for many areas in mathematics and applied sciences.

Linear algebra simplifies complex relationships through matrices and vectors. It's commonly used in:

• Computer science for algorithms, especially in machine learning and data science.
• Engineering for control systems and circuit analysis.
• Physics for formulating physical models and simulations.
• Economics for modeling economic theories and outcomes.

A core aspect of linear algebra is solving systems of linear equations. These systems can be represented as a single matrix equation, making calculations more efficient. Understanding linear algebra opens doors to more advanced mathematical concepts like eigenvectors and eigenvalues, used extensively in various fields.

2x2 Matrix

A 2x2 matrix is a square matrix with two rows and two columns. It is one of the simplest forms of matrices used in linear algebra. Despite its simplicity, a 2x2 matrix is powerful and can represent a wide range of mathematical concepts.

The determinant of a 2x2 matrix is critical for understanding its properties, particularly its invertibility. The formula for the determinant \( \text{det}(A) \) of a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \).

If the determinant of a 2x2 matrix is non-zero, the matrix is invertible; otherwise, it is singular. Invertible matrices have an inverse, meaning that they can "undo" the effect of the transformation they represent. In contrast, singular matrices lack an inverse and require alternative methods for solving related mathematical problems.

Understanding 2x2 matrices helps in grasping more complex matrix operations and their applications in real-world problems, such as modeling computer graphics or analyzing statistical data in multivariable calculus.