Question

Find the determinant of the \(2 \times 2\) matrix. . \(\left[\begin{array}{cc}1 & -3 \\ 7 & 7\end{array}\right]\)

Short answer

The determinant of the matrix is 28.

Step-by-step solution

Understanding the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula \( ad - bc \). We need to identify the values of \(a\), \(b\), \(c\), and \(d\) from the matrix and plug them into this formula.

Identify the Elements of the Matrix

In the given matrix \( \begin{bmatrix} 1 & -3 \ 7 & 7 \end{bmatrix} \), the elements are \( a = 1 \), \( b = -3 \), \( c = 7 \), and \( d = 7 \).

Apply the Determinant Formula

Substitute the identified elements into the determinant formula: \( ad - bc \). This becomes: \( (1)(7) - (-3)(7) \).

Perform the Calculations

Calculate the expression: \( (1)(7) = 7 \), and \( (-3)(7) = -21 \). Therefore, the determinant is \( 7 - (-21) = 7 + 21 = 28 \).

Key concepts

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices used in linear algebra. It consists of two rows and two columns. The general form of a 2x2 matrix is \[\begin{bmatrix} a & b \c & d \end{bmatrix},\] where each letter represents an element within the matrix.

Understanding a 2x2 matrix involves recognizing these key components:

• Each position in the matrix has a specific name, such as 'a' for the top-left element, 'b' for the top-right, 'c' for the bottom-left, and 'd' for the bottom-right.
• These matrices are used in various calculations, including finding the determinant, which helps in understanding properties such as invertibility.

Learning to identify these elements is a stepping stone to mastering matrix algebra, as they form the basis of more complex operations.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices and the rules to manipulate them. It encompasses various operations such as addition, subtraction, multiplication, and finding determinants. Each operation follows specific rules:

• Addition and Subtraction: Matrices can be added or subtracted element-wise, provided they are of the same dimensions.
• Multiplication: Unlike numbers, matrices have a different multiplication process. It involves taking the dot product of the rows and columns of two matrices. Not all matrices can be multiplied together.
• Determinants: The determinant finds the scalar value that can indicate certain properties like the invertibility of the matrix. For a 2x2 matrix, it's calculated using the formula \( ad - bc \) as seen in the given exercise.

Mastery of these operations opens up the path to solving complex problems in linear algebra.

Linear Algebra

Linear algebra is a fundamental field of mathematics focusing on vector spaces and linear mappings between these spaces. It uses matrices as a crucial tool to solve systems of linear equations. The concepts explored in matrix algebra form a core part of linear algebra studies.

Key components of linear algebra include:

• Vector Spaces: These are mathematical structures formed by vectors that can be added together and multiplied by numbers. They provide the framework for matrix operations.
• Linear Transformations: This involves taking one set of coordinates and transforming them into another, primarily using matrices, capturing a wide range of applications in geometry and physics.
• Solving Systems of Equations: Linear algebra provides methods for using matrices to find solutions to systems of equations, often through row reduction or matrix inverses.

Ultimately, mastery of linear algebra and matrix algebra allows for effective problem-solving in many scientific and engineering contexts.