Question

The sum of the eigen values of \(A\) equals to the trace of \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 3 & -1 \\ 0 & -1 & 3\end{array}\right]\).

Short answer

The sum of the eigenvalues of matrix \(A\) is 7.

Step-by-step solution

Definition of Trace

The trace of a matrix is defined as the sum of the elements on the main diagonal of the matrix. This is a key property related to eigenvalues.

Identify the Main Diagonal

For the matrix \(A\), identify the main diagonal elements. The given matrix \(A\) is \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 3 & -1 \ 0 & -1 & 3 \end{bmatrix}\), so the diagonal elements are \(1\), \(3\), and \(3\).

Calculate the Trace

Add the elements of the main diagonal together to find the trace. Here, the trace \(\text{tr}(A)\) is \(1 + 3 + 3 = 7\).

Sum of Eigenvalues Equal to Trace

By the property of matrices, the sum of the eigenvalues of a matrix is equal to its trace. Therefore, the sum of the eigenvalues of the matrix \(A\) is 7.

Key concepts

Trace of a Matrix

The trace of a matrix is a simple yet important concept in linear algebra. It is obtained by summing up the elements that are located on the main diagonal of a square matrix. The main diagonal runs from the top left to the bottom right of the matrix. To actually find the trace, you look at each entry on this diagonal and perform a straightforward summation.

For example, consider a matrix \( A \):

• The diagonal elements are the ones such as \(a_{11}, a_{22}, a_{33},...\) for an \(n \times n\) matrix which ensures they line up in a slanting line going down through the matrix.
• The trace is denoted as \( \text{tr}(A) \).
• A useful property is that the trace is only defined for square matrices since only these have a main diagonal that runs continuously from one corner to the opposite.
Summing these values gives us insight into other matrix properties, like the sum of its eigenvalues.

Hence in matrix \(A\), with diagonal elements \(1, 3, \) and \(3\), the trace is calculated as \(1 + 3 + 3 = 7\).

Main Diagonal of a Matrix

The main diagonal of a matrix is a crucial concept when discussing matrices, particularly in the context of understanding the trace and other properties. The main diagonal is defined as the set of elements that span from the top-left corner to the bottom-right corner of a square matrix.

Understanding this concept is very straightforward but important:

• Only square matrices possess a main diagonal.
• In a 3x3 matrix like \(A\), the elements are \(a_{11}, a_{22}, a_{33}\).
• Each diagonal element has the same row and column index.

Why is the main diagonal important? Because it is used in numerous calculations, such as determining the trace of a matrix or finding matrix invariants like eigenvalues. For matrix \(A\), identifying the main diagonal, helps us calculate its trace and find the eigenvalue sum. Essentially, the main diagonal simplifies mathematical operations involving matrices, serving as a foundation block for more complex concepts.

Properties of Matrices

Matrices have unique properties that play a fundamental role in various fields such as physics, computer science, and engineering. These properties are especially relevant when dealing with eigenvalues, as these matrix characteristics help establish solutions to systems of linear equations or transformations. Here are some key properties to keep in mind:

• Commutativity: Generally, matrix multiplication is not commutative, meaning \(AB eq BA\). However, specific matrices may be an exception under certain conditions.
• Associativity: Matrices follow associative property in multiplication, so \( (AB)C = A(BC) \).
• Identity Matrix: An identity matrix does not change a matrix when multiplied, forming the foundation for determining inverses.
• Inverse Matrix: When a matrix is multiplied by its inverse, the result is the identity matrix. Not all matrices have inverses, however.
• Determinants: Only square matrices have determinants, which provide information about the matrix's invertibility and eigenvalues.

Recognizing these properties equips you with the tools needed to manipulate and understand matrices more deeply, helping you solve complex problems. One key insight is that a matrix's eigenvalues are directly related to its trace, showcasing the elegant interplay among matrix characteristics.