Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{lllll} 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 120.

Step 1: Recognize the Matrix Type

Recognize that the matrix is a diagonal matrix. A square matrix is a diagonal matrix if all its off-diagonal elements are zero. That is, for a matrix to be diagonal, we must have \(a_{ij}=0\) for \(i\neq j\). In this exercise, the given matrix is a 5x5 diagonal matrix.

Step 2: Calculate the Determinant

The determinant of a diagonal matrix is the product of its diagonal elements. That is, if \(D=[a_{ij}]\) is a diagonal matrix, then \(det(D)=a_{11}*a_{22}*a_{33}*...*a_{nn}\). Therefore the determinant of the given matrix is the product of its diagonal elements i.e., \(det(D)=1*2*3*4*5=120\).

Key Concepts

Diagonal Matrices

In the world of linear algebra, a diagonal matrix is a fairly straightforward concept, yet it holds great importance. A diagonal matrix is defined as a square matrix where all the elements outside the main diagonal are zero.
For example, given a matrix of size 5x5, like the one in our exercise, it has non-zero elements only along the diagonal that runs from the top-left to the bottom-right.

• The main diagonal consists of elements like 1, 2, 3, 4, 5 in our given matrix.
• All other elements in the matrix, those which are not on this diagonal, are zeros.
• This structure makes operations like finding the determinant very straightforward.

Understanding diagonal matrices is crucial because they simplify many matrix operations. In particular, calculating determinants and eigenvalues becomes incredibly easy when dealing with diagonal matrices.

Matrix Properties

Matrices come with a set of properties that make them versatile tools in mathematics. Specifically, diagonal matrices have unique properties that ease calculations.
One of the defining properties of a diagonal matrix is how its determinant is calculated. The determinant of such a matrix is simply the product of the numbers on its main diagonal.

• For example, for our 5x5 matrix, the determinant is calculated as: \(1 \times 2 \times 3 \times 4 \times 5 = 120\).
• This property reduces the complexity significantly as there is no need to perform additional row or column operations.
• This simplicity is because the off-diagonal elements do not affect the determinant.

This property makes diagonal matrices particularly useful in solving linear equations and transforming systems without changing their determinant, a feature commonly used in various applications of linear transformations.

Linear Algebra Concepts

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It provides the foundation for many practical applications such as computer graphics, machine learning, and engineering.
One of the core concepts within linear algebra is understanding different types of matrices, such as diagonal matrices, and knowing their properties. This helps to swiftly solve problems and expand mathematical functionality.

• Linear transformations, often represented by matrices, help transform inputs across different dimensions and spaces.
• Concepts like matrix multiplication, determinants, and inversions stem from understanding matrix types.
• This comprehension gives you the tools necessary to manipulate spaces and functions effectively.

Overall, mastering these concepts gives you a profound insight into not just theoretical aspects but also real-world applications, where matrices and linear transformations are continuously at play.