Problem 9 - Advanced Engineering Mathematics

Question

In Problems 9-14, evaluate the determinant of the given matrix. $$ (-7) $$

Step-by-Step Solution

Answer

The determinant is -7.

Step 1: Understand the Matrix

Identify the matrix from the problem. Here, the matrix is represented by a single element, \[A = (-7)\] which is a 1x1 matrix.

Step 2: Evaluate the Determinant of a 1x1 Matrix

For a 1x1 matrix, the determinant is the value of the single element itself. Therefore, the determinant of matrix \( A \) is \(-7\).

Key Concepts

1x1 matrix

A 1x1 matrix is the simplest type of matrix, consisting of only one element. Imagine a number sitting alone in a grid – that's your 1x1 matrix. In mathematical notation, a 1x1 matrix is represented by placing a single number inside square brackets, for example \[\begin{bmatrix}a\\end{bmatrix}\] where \(a\) is a number. Because there’s only one element, there’s no need to perform complex calculations. Understanding this simple matrix sets the stage for grasping more complex matrices later on.
The single element in this matrix is both a row and a column by itself. Therefore, when you hear about a 1x1 matrix, think of it as a basic building block in matrix algebra. This simplicity makes it easy and straightforward to work with.

evaluating determinants

Evaluating determinants is a fundamental concept in linear algebra. When you encounter a 1x1 matrix, the process of finding the determinant is particularly straightforward. The determinant of a 1x1 matrix is simply the value of its lone element. For example, to find the determinant of \[\begin{bmatrix}-7\\end{bmatrix} \]one merely needs to take the single value, which in this case is \(-7\).
This simplicity, however, belies the determinant's usefulness. Determinants provide essential insights into certain properties of matrices, like whether a matrix is invertible. When the determinant is zero, the matrix doesn’t have an inverse. Thankfully, with a 1x1 matrix, evaluating determinants couldn't be easier.
For more complex matrices, the process involves breaking down the matrix into smaller parts, often using minor determinants, but keep the principle of evaluating the value in mind for solving any size matrix.

matrix algebra

Matrix algebra is the branch of mathematics involving matrices and the operations that can be performed on them. Understanding 1x1 matrices is a gateway to this subject as they provide the most basic form demanding minimal computation. In matrix algebra, one can perform operations like addition, subtraction, and multiplication of matrices.
A key operation is finding the determinant, which is crucial for solving system equations, performing transformations, and more. For a matrix to be invertible, its determinant must be non-zero; hence why the determinant is a critical concept in matrix algebra.
Matrix algebra helps us perform calculations that simplify working with multiple variables and systems of equations, prevalent in fields like engineering, computer science, and economics. As you move into larger matrices, you’ll employ rules of matrix algebra to determine properties and solve matrix-related problems. Begin with understanding the foundation, which includes simple 1x1 matrices, and you'll find yourself well-equipped to handle more complex algebraic tasks.