Problem 41 - Algebra and Trigonometry

Question

Evaluate each determinant. $$\left|\begin{array}{rr}{\left|\begin{array}{rr}{3} & {1} \\\\{-2} & {3}\end{array}\right|} & {\left|\begin{array}{rr}{7} & {0} \\\\{1} & {5}\end{array}\right|} \\\\{\left|\begin{array}{rr}{3} & {0} \\\\{0} & {7}\end{array}\right|} & {\left|\begin{array}{rr}{9} & {-6} \\\\{3} & {5}\end{array}\right|}\end{array}\right|$$

Step-by-Step Solution

Answer

Follow the above steps to find the determinants of each 2x2 matrix, and use these values to find the determinant of the larger 2x2 matrix. Square matrices are used to represent systems of linear equations, so understanding determinants is crucial for solving such systems.

Step 1: Compute the determinant of the smaller matrices

For each of the following matrices, compute the determinant, denoted as \(det\) or |\ | :\[\begin{array} {11}{3} & {1} \\{-2} &{3}\end{array}\]\[\begin{array} {11}{7} & {0} \\{1} &{5}\end{array}\]\[\begin{array} {11}{3} & {0} \\{0} &{7}\end{array}\]\[\begin{array} {11}{9} & {-6} \\{3} &{5}\end{array}\]Calculate the determinants by subtracting the product of the off-diagonal elements from the product of the diagonal elements, according to the following formula \( det(A) = a*d - b*c \), where \( a \), \( b \), \( c \) and \( d \) are the components of the $2x2$ matrix.

Step 2: Substitute the determinants into the larger 2x2 matrix and compute the determinant

Substitute the smaller determinants into the larger 2x2 matrix. Afterward, the determinant of the larger 2x2 matrix should be calculated using the same method as described in step 1.

Step 3: Solve the exercise

The determinant calculated in Step 2 is the solution to the exercise.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the study and manipulation of matrices. Matrices are rectangular arrays of numbers, symbols, or expressions, organized in rows and columns. They are essential in various fields, including physics, computer science, and engineering.
Matrix algebra involves operations like addition, subtraction, and multiplication of matrices, which follow specific rules. One of the key operations is finding the determinant; a special scalar value that provides a lot of insights into the matrix.

• Determinants: Used in solving linear equations, finding inverse matrices, and detecting if a matrix has full rank.
• Matrix Operations: Help solve complex problems involving systems of equations and transformations.
• Applications: Commonly used in statistics, economics, and computer graphics.

Understanding matrix algebra is fundamental to working efficiently with mathematical models and computations in various disciplines.

2x2 Matrices

2x2 matrices are the simplest kind of square matrix, consisting of two rows and two columns. Despite their simplicity, they are incredibly useful in linear algebra and have a wide range of applications.
A typical 2x2 matrix is represented as:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]In this context, the matrix contains four elements: \(a\), \(b\), \(c\), and \(d\).

• Simplicity: Easy to handle computation processes like determinants and inverses.
• Applications: Used to model transformations in 2-dimensional space, such as rotations and reflections.
• Determinants: For a 2x2 matrix, the determinant is \(ad - bc\), a simple calculation that offers a lot of information about the matrix.

Their structure makes 2x2 matrices perfect starters for students to grasp the fundamentals of matrix algebra.

Linear Algebra

Linear algebra is the field of mathematics concerning vector spaces and linear mappings between them. It involves the study of lines, planes, and subspaces, but more broadly, it looks at operations on vector spaces.
One of the primary tools in linear algebra is the matrix, which helps simplify the representation of complex linear systems, making them more manageable to solve or analyze.

• Vector Spaces: Fundamental to linear algebra, providing a setting for many algebraic operations.
• Linear Equations: Often represented and solved using matrices and determinants.
• Foundations: Vital in developing algorithms in computer science, machine learning, and data analysis.

Linear algebra's concepts such as eigenvalues and eigenvectors also have enormous significances, like in stability analysis and in transforming data for machine learning tasks.

Matrix Operations

Matrix operations are mathematical procedures that can be performed on matrices. These include addition, subtraction, scalar multiplication, and multiplication of matrices. Each operation has its own rules, especially multiplication, which is not as straightforward as number multiplication.
To multiply matrices, the number of columns in the first matrix must match the number of rows in the second. The resulting matrix facilitates solving multiple linear equations simultaneously, through methods like the elimination or inversion.

• Determinant Computation: Involves evaluating a specific value that helps in understanding properties like invertibility and singularity of matrices.
• Inversion: Finding an inverse involves using determinants, and it's applicable when solving complex system of equations.
• Complex Systems: Supports the handling and manipulation of large data sets and systems of linear equations.

Mastering matrix operations is pivotal for fields requiring extensive numerical computations, such as physics simulations, 3D modeling, and predictions in economics.