Question

Find the trace of the given matrix. \(\left[\begin{array}{ccc}-2 & -3 & 5 \\ 5 & 2 & 0 \\ -1 & -3 & 1\end{array}\right]\)

Short answer

The trace of the matrix is 1.

Step-by-step solution

Understanding the Trace of a Matrix

The trace of a matrix is the sum of the elements on its main diagonal. The main diagonal of a square matrix starts from the top left corner and ends at the bottom right corner.

Identifying the Main Diagonal

Given the matrix \[\begin{bmatrix}-2 & -3 & 5 \5 & 2 & 0 \-1 & -3 & 1\end{bmatrix}\]The elements on the main diagonal are \(-2\), \(2\), and \(1\).

Calculating the Trace

To find the trace, sum the elements on the main diagonal. Compute \(-2 + 2 + 1\).

Performing the Calculation

The calculation is: \[-2 + 2 + 1 = 1\]So, the trace of the matrix is \(1\).

Key concepts

Main Diagonal

When talking about matrices, the main diagonal is a key feature to understand. A main diagonal is found in a square matrix, meaning the number of rows is equal to the number of columns. It runs diagonally across the matrix from the top-left corner to the bottom-right corner. Let's visualize it: given a 3x3 matrix, you would count elements like this: the first element of the first row, second element of the second row, and third element of the third row.
In our original exercise, the elements \(-2, 2, \) and \(1\) are on the main diagonal of the matrix:

• The first main diagonal element is \(-2\).
• The second main diagonal element is \(2\).
• The third main diagonal element is \(1\).

This path of elements is vital in matrix operations like finding the trace.

Matrix Algebra

Matrix algebra is a branch of mathematics working with matrices to solve system equations, transform geometric data, and more. A matrix is essentially a rectangular array of numbers arranged in rows and columns. Understanding operations like addition and multiplication is crucial in matrix algebra.
Addition involves taking two matrices of the same dimension and simply adding their corresponding entries. Multiplication is more complex; the number of columns of the first matrix must be the same as the number of rows of the second matrix. A popular operation in matrix algebra is finding the trace, especially important in specific matrix calculations or transformations.
Through operations like these, matrix algebra provides a powerful framework for working with complex data efficiently.

Sum of Diagonal Elements

The sum of the diagonal elements of a matrix has a special name in linear algebra — it's called the trace. The trace is simply the sum of all the elements found on a matrix's main diagonal. It's straightforward but holds considerable significance in many mathematical and physical applications.
\[\text{Trace} = a_{11} + a_{22} + \ldots + a_{nn}\]
Where \(a_{11}, a_{22}, \ldots, a_{nn}\) are the main diagonal elements. In the exercise given, to find the trace of \(\begin{bmatrix}-2 & -3 & 5 \ 5 & 2 & 0 \ -1 & -3 & 1\end{bmatrix}\), sum the main diagonal elements:

• \(-2 + 2 + 1 = 1\)

This simple addition provides the trace of the matrix, summarizing key aspects of its structure or transformations.