Problem 21 - College Algebra with Modeling and Visualization

Question

Let \(A\) be the given matrix. Use technology to calculate det \(A\). $$ \left[\begin{array}{rr} 11 & -32 \\ 1.2 & 55 \end{array}\right] $$

Step-by-Step Solution

Answer

det \(A = 643.4\).

Step 1: Determine the Formula for the Determinant of a 2x2 Matrix

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

Step 2: Substitute the Matrix Values into the Formula

For the given matrix \(A = \begin{bmatrix} 11 & -32 \ 1.2 & 55 \end{bmatrix}\), substitute the values into the formula: \(a = 11\), \(b = -32\), \(c = 1.2\), and \(d = 55\).

Step 3: Calculate the Product of the Diagonal Elements

Calculate \(ad = 11 \times 55 = 605\). This is the product of the main diagonal elements of the matrix.

Step 4: Calculate the Product of the Off-Diagonal Elements

Calculate \(bc = -32 \times 1.2 = -38.4\). This is the product of the off-diagonal elements of the matrix.

Step 5: Compute the Determinant Using the Formula

Compute the determinant using the expression \(\text{det}(A) = ad - bc = 605 - (-38.4) = 605 + 38.4 = 643.4\).

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of matrices used in mathematics, especially in linear algebra. It consists of two rows and two columns. A general form of a 2x2 matrix can be represented as: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix.
2x2 matrices are fundamental in performing various operations like calculating determinants, which come in handy in characterizing the properties of the matrix, such as invertibility.

• The determinant provides insight into the matrix's ability to transform another matrix.
• An identity matrix, which is a special 2x2 matrix, looks like \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), and it serves as the multiplication identity in matrix operations.

Mathematical tools often leverage 2x2 matrices due to their simplicity, yet they capture basic concepts needed for understanding more complex matrices.

matrix operations

Matrix operations include actions like addition, subtraction, and multiplication. Each operation has its own set of rules. These operations are important for solving linear equations, transformations, and other tasks in linear algebra.

• Addition and Subtraction: Two matrices can be added or subtracted only if they have the same dimensions. The operation is done by adding or subtracting corresponding elements of the matrices.
• Multiplication: Matrix multiplication is a bit more complex. For multiplication, the number of columns in the first matrix must equal the number of rows in the second one. This operation involves taking the dot product of rows and columns.

Understanding these operations is pivotal because they form the basis for more advanced concepts such as eigenvalues and eigenvectors. Additionally, the determinant, particularly for a 2x2 matrix, is defined primarily using the multiplication of its elements as illustrated in the formula \( \text{det}(A) = ad - bc \). Keeping these basic operations in mind helps in grasping how matrices function in various mathematical computations.

linear algebra

Linear algebra is a field of mathematics that revolves around the study of vectors, vector spaces, and linear transformations.
It provides the tools for manipulating linear equations and modeling multi-variable systems.
In linear algebra, matrices like the 2x2 matrix are used to represent linear transformations and systems of linear equations.

• Determinants: In linear algebra, the determinant is a scalar value that is a function of the entries of a square matrix. It provides important information about the matrix, including whether it is invertible.
• Eigenvalues and Eigenvectors: These concepts assess the magnitude and direction of matrix transformations, respectively. They are derived from solving particular equations involving the matrix for which the determinants play a crucial role.

Linear algebra is not only theoretical but has practical applications in science, engineering, and computer science. It helps in understanding systems and solving equations that model real-world scenarios. Grasping these key concepts allows you to understand how linear algebra applies across different domains.