Problem 12
Question
Determine \(\operatorname{tr}(A)\) for the given matrix. $$A=\left[\begin{array}{lll} 2 & 0 & 1 \\ 3 & 2 & 5 \\ 0 & 1 & -5 \end{array}\right]$$
Step-by-Step Solution
Verified Answer
The trace of the given matrix A is the sum of its diagonal elements: \(2 + 2 + (-5) = -1\). Therefore, \(\operatorname{tr}(A) = -1\).
1Step 1: Identify the diagonal elements
The diagonal elements of matrix A are the ones located in the positions (1,1), (2,2), and (3,3). For our given matrix A:
\[A=\left[\begin{array}{lll}
2 & 0 & 1 \\\
3 & 2 & 5 \\\
0 & 1 & -5
\end{array}\right]\]
The diagonal elements are 2, 2, and -5.
2Step 2: Calculate the sum of the diagonal elements
Now that we have identified the diagonal elements of matrix A, we will calculate their sum to find the trace of A.
The sum of the diagonal elements is:
\[2 + 2 + (-5)\]
3Step 3: Simplify the sum
Simplify the sum of the diagonal elements:
\[2 + 2 - 5 = 4 - 5\]
4Step 4: Find the trace of matrix A
The trace of matrix A is the simplified sum of its diagonal elements:
\[\operatorname{tr}(A) = 4 - 5 = -1\]
So, the trace of the given matrix A is -1.
Key Concepts
Diagonal ElementsMatrix AdditionSquare Matrix
Diagonal Elements
When dealing with matrices, identifying the diagonal elements is an essential step, especially when you need to find the trace.
Diagonal elements are those positioned along the 'main diagonal' of a square matrix. In more technical terms, these elements are located at the intersection of equal row and column indices, such as positions (1,1), (2,2), (3,3), and so on.
When we look at matrix A:
\[A=\begin{bmatrix}2 & 0 & 1 \3 & 2 & 5 \0 & 1 & -5\end{bmatrix}\]We see that the diagonal elements are:
Diagonal elements are those positioned along the 'main diagonal' of a square matrix. In more technical terms, these elements are located at the intersection of equal row and column indices, such as positions (1,1), (2,2), (3,3), and so on.
When we look at matrix A:
\[A=\begin{bmatrix}2 & 0 & 1 \3 & 2 & 5 \0 & 1 & -5\end{bmatrix}\]We see that the diagonal elements are:
- 2 at position (1,1)
- 2 at position (2,2)
- -5 at position (3,3)
Matrix Addition
Matrix Addition is a fundamental operation where two matrices of the same size are added together by summing their corresponding elements.
However, in the context of calculating the trace, we are specifically interested in adding together only the diagonal elements, as demonstrated in the original exercise.
Let's revisit matrix A for a quick clarification:
\[A = \begin{bmatrix}2 & 0 & 1 \3 & 2 & 5 \0 & 1 & -5\end{bmatrix}\]To find the trace, we only need to add the diagonal elements: 2, 2, and -5. This addition gives us the trace of the matrix:
However, in the context of calculating the trace, we are specifically interested in adding together only the diagonal elements, as demonstrated in the original exercise.
Let's revisit matrix A for a quick clarification:
\[A = \begin{bmatrix}2 & 0 & 1 \3 & 2 & 5 \0 & 1 & -5\end{bmatrix}\]To find the trace, we only need to add the diagonal elements: 2, 2, and -5. This addition gives us the trace of the matrix:
- The sum is calculated as \(2 + 2 + (-5)\)
- Simplifying, it becomes 4 - 5 = -1
Square Matrix
A Square Matrix is a type of matrix where the number of rows equals the number of columns. The structure of a square matrix makes it suitable for various mathematical operations, including the calculation of the trace.
The matrix A presented in the exercise is a 3x3 square matrix:
\[A = \begin{bmatrix}2 & 0 & 1 \3 & 2 & 5 \0 & 1 & -5\end{bmatrix}\]For this square matrix, with equal dimensions, identifying diagonal elements becomes straightforward.
The presence of these equal dimensions allows us to comfortably perform operations like finding the trace, which involves adding up the diagonal elements.
Hence, understanding square matrices is integral to comprehending why the trace operation is possible and how it operates within such structures.
The matrix A presented in the exercise is a 3x3 square matrix:
\[A = \begin{bmatrix}2 & 0 & 1 \3 & 2 & 5 \0 & 1 & -5\end{bmatrix}\]For this square matrix, with equal dimensions, identifying diagonal elements becomes straightforward.
The presence of these equal dimensions allows us to comfortably perform operations like finding the trace, which involves adding up the diagonal elements.
Hence, understanding square matrices is integral to comprehending why the trace operation is possible and how it operates within such structures.
Other exercises in this chapter
Problem 12
Use Gaussian elimination to determine the solution set to the given system. $$\begin{aligned} x_{1}+2 x_{2}+x_{3}+x_{4}-2 x_{5} &=3 \\ x_{3}+4 x_{4}-3 x_{5} &=2
View solution Problem 12
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{
View solution Problem 13
Let \(A\) and \(B\) be \(n \times n\) matrices. If \(A\) is skew-symmetric use properties of the transpose to establish that \(B^{T} A B\) also is skew- symmetr
View solution Problem 13
Express the matrix \(A\) as a product of elementary matrices. $$A=\left[\begin{array}{lll}1 & 2 & 3 \\\0 & 8 & 0 \\\3 & 4 & 5\end{array}\right]$$
View solution