Question

Find the determinant of the matrix. $$ [6] $$

Short answer

The determinant of the given matrix is 6.

Step-by-step solution

Identify the matrix

The given matrix is a 1x1 matrix, also known as a scalar. It only contains one element, which is 6.

Find the determinant of the scalar

The determinant of a scalar is the scalar itself. Since our matrix only contains one element (6), the determinant of this matrix is 6.

Key concepts

1x1 Matrix

A 1x1 matrix is the most basic form of a matrix and is quite unique. It contains exactly one element. Because of this, it can be viewed as both a matrix and a single number, often referred to as a scalar. A 1x1 matrix is represented simply as \[[ a ]\]where 'a' is the single element within the matrix. The simplicity of a 1x1 matrix means that many matrix operations become much simpler compared to larger matrices. Understanding this concept is essential before diving into more complex matrix operations that involve larger matrices.

Scalar

A scalar is a single number, and in the context of matrices, it refers to a matrix with only one element, the 1x1 matrix. Scalars are fundamental in mathematics because they represent quantities without direction, unlike vectors or matrices that hold more complex data. For example:

• The number 6 can be seen as a scalar.
• When in a 1x1 matrix, it looks like this: \[[6]\]

In operations involving scalars, many mathematical rules become straightforward. For instance, the determinant of a scalar, which is essentially a 1x1 matrix, is the number itself. Thus, if you have a matrix like \[[6]\]it's simple to find that the determinant is 6.

Matrix Operations

Matrix operations can vary from simple to complex. With a 1x1 matrix, these operations are simplified significantly. Let's explore some basic operations:- **Addition and Subtraction**: If you add or subtract two 1x1 matrices, it is simply the addition or subtraction of their respective elements. For example, \[ [6] + [2] = [8] \]- **Multiplication**: Multiplying a 1x1 matrix by another matrix (1x1) results in a matrix where the single element is the product of the two numbers. For example: \[ [6] \times [2] = [12] \]- **Determinant**: As illustrated, the determinant of a 1x1 matrix is the scalar itself. In practical terms, if the matrix is \[[6]\]the determinant is 6.Though 1x1 matrices might seem overly simplistic, they create a foundation for understanding more expansive matrix operations and serve as building blocks for larger, more complex matrices.