Question

Find the determinant of the matrix. $$ [-5] $$

Short answer

-5

Step-by-step solution

Determinant of a 1x1 Matrix

Determining the determinant of a 1x1 matrix is straightforward because the determinant of a 1x1 matrix containing the element a is just a itself. In this case, the matrix has one element, which is -5. Therefore, the determinant of this matrix is -5.

Key concepts

1x1 Matrix

Understanding a matrix begins with the simplest form—a 1x1 matrix. This type of matrix is as elementary as it gets, consisting of a single number. In linear algebra, this single element can represent a scalar quantity, a simple numeric value with no other dimensions or directions to consider.

A 1x1 matrix, such as the \[ -5 \] given in our exercise, is essentially an encapsulated version of the number itself. When we talk about finding its determinant, we are looking for a value that can tell us something about the transformation the matrix represents. For 1x1 matrices, this transformation is quite straightforward; it either enlarges or shrinks the magnitude depending on the number, without any direction since there is no concept of dimensions beyond the single point.

The determinant of a 1x1 matrix is critical, albeit simple, as it forms the building block for understanding larger, more complex matrices. It's the starting point for anyone delving into matrix algebra and linear algebra.

Matrix Algebra

Matrix algebra is a specialized branch of mathematics that deals with the study and manipulation of matrices—arrays of numbers arranged in rows and columns. In this context, matrices are not just static arrays of data; they are dynamic entities that encode information about linear transformations and can be combined and manipulated using specific rules.

Operations in matrix algebra include addition, subtraction, multiplication, and finding inverses and determinants. The determinant is a unique number associated with a square matrix that can tell us many important characteristics about the matrix, such as whether it is invertible or the volume scale factor of the linear transformation that the matrix represents.

Importance of Determinant in Matrix Algebra

In the context of matrix algebra, the determinant plays a significant role. It is often used to solve systems of linear equations, to find the inverse of a matrix, and to determine properties of the matrix such as singularity. The determinant reflects how a linear transformation will scale areas or volumes; if the determinant is zero, the transformation squashes space into a lower dimension, and the matrix does not have an inverse.

Linear Algebra

Linear algebra is a field of mathematics that is concerned with vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. The essence of linear algebra lies in how it can represent and solve complex problems in a highly systematic way using matrices and vectors.

With linear algebra, you can describe many real-world phenomena, such as physics simulations, computer graphics, engineering problems, and economic models, with relative ease. It's also the bedrock for more advanced topics in math, science, and engineering, like quantum mechanics, machine learning, and optimization.

Linear Transformations and Determinants

Central to linear algebra is the concept of linear transformations—functions that map vectors from one vector space to another while preserving operations of vector addition and scalar multiplication. Matrices are used to represent these linear transformations, and determinants measure how the transformation affects volume. When you calculate the determinant of a matrix, you're essentially finding out whether the space is being stretched or compressed, and by how much, which has countless applications in theoretical and applied disciplines.