Question

Find the trace of the given matrix. \(\left[\begin{array}{cccc}5 & 2 & 2 & 2 \\ -7 & 4 & -7 & -3 \\ 9 & -9 & -7 & 2 \\ -4 & 8 & -8 & -2\end{array}\right]\)

Short answer

The trace of the matrix is 0.

Step-by-step solution

Understanding the Trace

The trace of a matrix is the sum of the elements on its main diagonal. In a square matrix, this is the diagonal that runs from the top left corner to the bottom right corner. For our 4x4 matrix, we need to identify the elements at positions (1,1), (2,2), (3,3), and (4,4).

Identify Diagonal Elements

Now identify the elements on the main diagonal of the given matrix: - First row, first column: 5. - Second row, second column: 4. - Third row, third column: -7. - Fourth row, fourth column: -2.

Calculate the Trace

Add the diagonal elements identified in the previous step:\[ 5 + 4 + (-7) + (-2) = 0 \]This is the sum of the elements on the main diagonal, which gives us the trace of the matrix.

Key concepts

Diagonal Elements

Diagonal elements in a matrix refer to those elements that are situated on the main diagonal. The main diagonal is what you get when you draw an imaginary line from the top left corner of a matrix to the bottom right corner.

So, why are diagonal elements important? In matrices, both for small and large, they often contain crucial information about the matrix itself. Whether you’re calculating a trace or working on more advanced operations like determinants, diagonal elements play a fundamental role.

In our exercise, we had a 4x4 matrix, and the diagonal elements were identified as 5, 4, -7, and -2. This selection was straightforward: each element is picked from the same row and column index (e.g., the element from row 1, column 1 is 5). Understanding the layout and order of these numbers is essential for matrix operations.

Square Matrix

A square matrix is a type of matrix where the number of rows is equal to the number of columns. This symmetry is what makes many operations, like finding the trace, possible.

For instance, if you're dealing with a 4x4 matrix, it means there are 4 rows and 4 columns. The matrix given in our exercise was indeed a 4x4 square matrix, which allowed us to properly examine its main diagonal.

Square matrices are special because they often open the door to interesting mathematical properties. Operations like determinant evaluation, eigenvalue computation, and finding the inverse are all feasible due to their square nature. Whenever dealing with a given matrix, first determining if it's square helps guide you to the right operations.

Trace Calculation

The trace of a matrix is a specific value calculated by summing all the diagonal elements from a square matrix. Not only is it simple, but it's powerful for certain applications, providing insights into matrix properties.

Calculating the trace starts by identifying the diagonal elements. As we saw in the provided solution steps, these were taken one by one. In our example, we had the elements 5, 4, -7, and -2. After identifying these, we simply add them up: \[ 5 + 4 + (-7) + (-2) = 0 \]

By adding them, we found that the trace of this matrix is 0. This process highlights just how important these elements are in determining specific characteristics of the matrix. Whether learned for basic class problems or more complex studies, tracing remains a vital computation.