Problem 11
Question
Determine \(\operatorname{tr}(A)\) for the given matrix. $$A=\left[\begin{array}{lll} 1 & 2 & -1 \\ 3 & 2 & -2 \\ 7 & 5 & -3 \end{array}\right]$$
Step-by-Step Solution
Verified Answer
The trace of matrix A is the sum of the main diagonal elements: \(1 + 2 - 3 = 0\). So, \(\operatorname{tr}(A) = 0\).
1Step 1: Identify the main diagonal elements
In matrix A, the main diagonal elements are the ones extending from the top-left corner to the bottom-right corner. They are: \(1\), \(2\), and \(-3\).
2Step 2: Compute the sum of the main diagonal elements
Now, we will add the main diagonal elements to determine the trace, so we have:
$$
\operatorname{tr}(A) = 1 + 2 + (-3)
$$
3Step 3: Calculate the trace
Lastly, let's calculate the sum:
$$
\operatorname{tr}(A) = 1 + 2 - 3 = 0
$$
The trace of matrix A is 0.
Key Concepts
Main Diagonal ElementsMatrix OperationsLinear Algebra
Main Diagonal Elements
The concept of the main diagonal elements is fundamental in various branches of mathematics, particularly in linear algebra. The main diagonal elements of a square matrix are the elements that extend from the top left corner of the matrix to the bottom right corner. These elements are crucial because they often hold special properties that are not shared by the other elements of the matrix.
For instance, when we analyze a matrix like
\[A=\begin{bmatrix} 1 & 2 & -1 \ 3 & 2 & -2 \ 7 & 5 & -3 \end{bmatrix}\],
the main diagonal elements are 1, 2, and -3. They form a kind of backbone for the matrix, often determining attributes of the matrix such as its trace, which is simply the sum of these diagonal elements.
Understanding the significance of the main diagonal is particularly important in exercises like evaluating the trace of a matrix, which leads to a deeper appreciation for how matrices can encode and simplify complex relationships within systems.
For instance, when we analyze a matrix like
\[A=\begin{bmatrix} 1 & 2 & -1 \ 3 & 2 & -2 \ 7 & 5 & -3 \end{bmatrix}\],
the main diagonal elements are 1, 2, and -3. They form a kind of backbone for the matrix, often determining attributes of the matrix such as its trace, which is simply the sum of these diagonal elements.
Understanding the significance of the main diagonal is particularly important in exercises like evaluating the trace of a matrix, which leads to a deeper appreciation for how matrices can encode and simplify complex relationships within systems.
Matrix Operations
Matrix operations are techniques or calculations that can be performed on matrices and include actions like addition, subtraction, multiplication, and the finding of the trace. These operations play a crucial role in linear algebra and have vast applications in various fields, including engineering, physics, computer science, and economics.
When we perform operations like finding the trace, we're usually looking for a characteristic value or property of a matrix. For example, the trace of a matrix, denoted \(\text{{tr}}(A)\), is a type of operation that combines elements of both addition and the recognition of specific positions (main diagonal) within the matrix. The process involves summing up only the main diagonal elements. It's a simpler operation when compared to others like multiplication, but still, it holds significant value in determining aspects like the eigenvalues of a matrix.
When we perform operations like finding the trace, we're usually looking for a characteristic value or property of a matrix. For example, the trace of a matrix, denoted \(\text{{tr}}(A)\), is a type of operation that combines elements of both addition and the recognition of specific positions (main diagonal) within the matrix. The process involves summing up only the main diagonal elements. It's a simpler operation when compared to others like multiplication, but still, it holds significant value in determining aspects like the eigenvalues of a matrix.
Linear Algebra
Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It is fundamental in understanding and working with multivariate systems, where mathematical structures known as matrices are employed to solve complex problems.
In the context of linear algebra, matrices are not merely collections of numbers; they represent linear transformations. A linear transformation can be visualized as moving, scaling, or rotating shapes without changing the essence of the shape itself. The trace, as we've identified in the example with matrix
\(A\), is an intrinsic value of a square matrix. It helps in the study of such transformations as it remains invariant under change of basis, making it a powerful tool in the study of linear equations and transformations. A zero trace, as in our exercise, might indicate a sort of 'balance' in the action of the transformation represented by the matrix.
In the context of linear algebra, matrices are not merely collections of numbers; they represent linear transformations. A linear transformation can be visualized as moving, scaling, or rotating shapes without changing the essence of the shape itself. The trace, as we've identified in the example with matrix
\(A\), is an intrinsic value of a square matrix. It helps in the study of such transformations as it remains invariant under change of basis, making it a powerful tool in the study of linear equations and transformations. A zero trace, as in our exercise, might indicate a sort of 'balance' in the action of the transformation represented by the matrix.
Other exercises in this chapter
Problem 11
Use Gaussian elimination to determine the solution set to the given system. $$\begin{aligned} x_{1}+2 x_{2}-x_{3}+x_{4} &=1 \\ 2 x_{1}-3 x_{2}+x_{3}-x_{4} &=2 \
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Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{
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Let \(A\) be an \(n \times n\) matrix. (a) Use the index form of the matrix product to write the \(i j\) th element of \(A^{2}.\) (b) In the case when \(A\) is
View solution Problem 12
Express the matrix \(A\) as a product of elementary matrices. $$A=\left[\begin{array}{rrr}0 & -4 & -2 \\\1 & -1 & 3 \\\\-2 & 2 & 2\end{array}\right]$$
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