Question

Find the trace of the given matrix. \(\left[\begin{array}{ccc}-4 & 1 & 1 \\ -2 & 0 & 0 \\ -1 & -2 & -5\end{array}\right]\)

Short answer

The trace of the matrix is -9.

Step-by-step solution

Understanding Trace

The trace of a matrix is the sum of its diagonal elements. For a 3x3 matrix, such as the one given, this means we will add the elements located at positions (1,1), (2,2), and (3,3).

Identify Diagonal Elements

For the given matrix \[\begin{bmatrix}-4 & 1 & 1 \-2 & 0 & 0 \-1 & -2 & -5\end{bmatrix}\] the diagonal elements are those that run from the top left to the bottom right: -4, 0, and -5.

Sum the Diagonal Elements

Add the identified diagonal elements together: \[-4 + 0 + (-5) = -4 - 5\].The sum of these elements is -9.

Key concepts

Diagonal Elements

In any square matrix, diagonal elements are those that form a line from the top left corner to the bottom right corner of the matrix. Let's break this down to understand the importance of these diagonal elements:

• The position of diagonal elements is what makes them unique. In a matrix with dimensions of 3x3, they are found at coordinates (1,1), (2,2), and (3,3).
• Diagonal elements possess significance in various mathematical operations, such as calculating the trace of a matrix.
• They often represent key features in applications, such as identity matrices where all diagonal elements are 1.

By clearly identifying these elements, complex matrix operations become simpler to manage.
This first step in understanding matrices leads to deeper comprehension as we apply these basics further.

3x3 Matrix

A 3x3 matrix refers to a square matrix with three rows and three columns. This size of matrix is commonly used in various areas of mathematics and engineering. Here are some notable points about 3x3 matrices:

• Due to their low dimensionality, 3x3 matrices are manageable and often serve as introductory examples for more advanced matrix concepts.
• The operations on 3x3 matrices, including addition, multiplication, or finding the determinant, are foundational skills in linear algebra.
• Such matrices can represent systems of equations, rotations in geometry, or data transformations in computer graphics.

Understanding a 3x3 matrix helps in building a comprehensive foundation for tackling larger matrices or engaging with complex systems in various scientific computations.

Sum of Diagonal Elements

Summing the diagonal elements of a matrix gives us what is known as the trace of the matrix. The process is straightforward, but remarkably powerful:

• To find the trace, you only need to add the elements at positions (i, i) for all i in an n x n matrix.
• In context with our 3x3 matrix, we look at the elements located at (1,1), (2,2), and (3,3).
• For our example matrix, the elements -4, 0, and -5 are summed to yield a trace of -9.
• The trace provides valuable insights for matrix-related properties, such as consistency checks or simplifying complex expressions.

Although simple, calculating the trace is a crucial technique widely used in theoretical and applied mathematics, underpinning various real-world applications in control systems, computer vision, and more.