Question

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

Short answer

The trace of the matrix is 10.

Step-by-step solution

Understand the Matrix

We are given a 3x3 matrix:\[\begin{bmatrix} 4 & 2 & -1 \ -4 & 1 & 4 \ 0 & -5 & 5\end{bmatrix} \]

Identify the Diagonal Elements

The diagonal elements of a matrix are located from the top left to the bottom right. For this matrix, the diagonal elements are 4, 1, and 5.

Calculate the Trace

The trace of a matrix is the sum of its diagonal elements. Here, we sum the diagonal elements: 4 + 1 + 5.

Sum Up the Values

Add the diagonal elements: \[4 + 1 + 5 = 10\]

Conclude the Solution

The trace of the matrix is 10.

Key concepts

trace of a matrix

The trace of a matrix is a fundamental concept in linear algebra that often appears in various mathematical contexts. It involves summing up all the elements on the main diagonal of a square matrix. A matrix's main diagonal runs from the top left corner to the bottom right corner. For example, in the provided 3x3 matrix, the diagonal elements are 4, 1, and 5.

To find the trace, you simply add these diagonal elements together:

• The first diagonal element is 4,
• The second diagonal element is 1,
• The third diagonal element is 5.

So, the trace is calculated as 4 + 1 + 5, which equals 10.
This calculation is quite straightforward, but understanding why the trace is important can be more interesting. The trace has properties that make it useful in areas such as differential equations and characterizing matrix behavior. For instance, in transformations, the trace gives insights into the matrix's scaling tendencies.

diagonal elements

The diagonal elements of a matrix are crucial in understanding its structure. They reside on the main diagonal, which stretches from the upper left to the lower right of the matrix. In any square matrix, this diagonal contains the same number of elements as the matrix's dimensions.

For the matrix presented:

• The diagonal includes the elements 4, 1, and 5.
• Each of these elements has a prominent role in various matrix operations, like calculating the trace.

The importance of diagonal elements extends beyond simple calculations. They can determine certain properties of matrices, such as whether a matrix is invertible or what the eigenvalues might resemble in a transformation.

In essence, while they may seem to be just entries in a row or column, diagonal elements hold significant sway over matrix characteristics, influencing both theoretical and practical applications in linear algebra.

matrix addition

Matrix addition might not directly affect the calculation of a trace, but understanding it can provide deeper insights into matrix operations. Matrix addition is a process where two matrices of the same dimensions are added together, element by element.

Suppose you have two matrices:

• Matrix A, where each element is added to the corresponding element in Matrix B.
• For instance, \[ \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}\]

In this example, each element is obtained by adding elements from the same position in the two matrices, illustrating the simplicity of matrix addition.
While not related directly to the trace, knowing how to perform matrix addition is essential for deeper topics in linear algebra, like solving systems of equations or transformations. This understanding encapsulates how matrices interact with each other and often underpins more advanced mathematical discussions.