Question

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{ll}-4 & -5 \\ -1 & -4\end{array}\right]\)

Short answer

The determinant is 11.

Step-by-step solution

Understand the Determinant Formula for a 2x2 Matrix

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

Identify Elements in the Matrix

Take the given matrix \( \begin{bmatrix} -4 & -5 \ -1 & -4 \end{bmatrix} \). Identify \( a = -4 \), \( b = -5 \), \( c = -1 \), and \( d = -4 \).

Apply the Determinant Formula

Substitute the values into the determinant formula: \( (-4)(-4) - (-5)(-1) \).

Calculate Each Part of the Expression

Calculate \( (-4) \times (-4) = 16 \) and \( (-5) \times (-1) = 5 \).

Perform the Subtraction for the Determinant

Subtract the results to find the determinant: \( 16 - 5 = 11 \).

Key concepts

2x2 matrix

When we talk about a 2x2 matrix, we refer to a small and compact matrix that consists of two rows and two columns. These matrices are particularly significant in basic linear algebra due to their simplicity and ease of manipulation. They are typically presented in the form:

• a b
• c d

A 2x2 matrix might look like this:\[\left[\begin{array}{cc}a & b \c & d\end{array}\right]\]Each element in the matrix has a specific position defined by its row and column, making it easy to identify and work with. For instance, in a matrix:

• a is the element in the first row, first column
• b is the element in the first row, second column
• c is the element in the second row, first column
• d is the element in the second row, second column

These elements are crucial for calculating the determinant or for performing various matrix operations. Recognizing how a 2x2 matrix is structured sets the foundation for delving deeper into understanding matrix arithmetic and other linear algebra concepts.

matrix operations

Matrix operations are mathematical procedures that involve manipulating matrices according to specific rules. Some of the basic operations include addition, subtraction, multiplication, and finding determinants.One important operation for 2x2 matrices is determinant calculation. The determinant provides a scalar value that can determine if the matrix has an inverse and gives insights into the matrix's properties. For a 2x2 matrix:\[\left[\begin{array}{cc}a & b \c & d\end{array}\right]\]the determinant is found using the formula:\[det(A) = ad - bc\]This operation involves multiplying across the diagonals, where 'a' and 'd' are multiplied together, and then subtracting the product of 'b' and 'c'. Here are the essential steps:

• Identify the elements in their respective positions.
• Multiply 'a' with 'd'.
• Multiply 'b' with 'c'.
• Subtract the second product from the first.

The result will be the determinant of the matrix, a vital value in further matrix analysis.

linear algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, and linear transformations, which can often be represented using matrices. It provides a framework for solving systems of linear equations, transformations, and understanding geometric concepts in higher dimensions. Central to linear algebra is the concept of a matrix, an array of numbers arranged in rows and columns used to represent linear transformations. Matrices allow us to encode a wealth of information and perform various operations to unravel it. In the realm of 2x2 matrices, these fundamental concepts are introduced step by step, helping students grasp the underlying principles of how linear spaces and transformations function, and how systems of equations can be represented and solved. Understanding linear algebra equips us to:

• Model real world data efficiently using equations and transformations.
• Solve linear systems using matrix methods.
• Understand vector spaces, eigenvalues, and eigenvectors.

It lays the groundwork for advanced mathematical topics and applications in physics, engineering, computer science, and statistics, driving innovations and solutions in numerous fields. Mastering basic operations on 2x2 matrices, like determining the determinant, acts as the starting point in exploring the captivating world of linear transformations and space.