Chapter 2: Problem 48
The sum and product of the eigen values of \(\left[\begin{array}{lll}2 & 2 & 1 \\\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right]\) are ......... and
Short Answer
Expert verified
Sum is 7, product is 5.
Step by step solution
01
Identify the Matrix
We are given the matrix \(A = \left[\begin{array}{lll}2 & 2 & 1 \ 1 & 3 & 1 \ 1 & 2 & 2\end{array}\right]\). We need to find the sum and product of its eigenvalues.
02
Use the Sum of Eigenvalues Property
The sum of the eigenvalues of a matrix equals the trace of the matrix. The trace of a matrix is the sum of its diagonal elements. Here, the diagonal elements of matrix \(A\) are 2, 3, and 2. Thus, the trace is \(2 + 3 + 2 = 7\). So, the sum of the eigenvalues is 7.
03
Use the Product of Eigenvalues Property
The product of the eigenvalues of a matrix equals the determinant of the matrix. We calculate the determinant of matrix \(A\) using the cofactor expansion method. The determinant is:\[\det(A) = 2\left((3)(2) - (1)(2)\right) - 2\left((1)(2) - (1)(1)\right) + 1\left((1)(2) - (1)(3)\right) = 2(6-2) - 2(2-1) + 1(2-3) = 2 \cdot 4 - 2 \cdot 1 + 1 \cdot (-1) = 8 - 2 - 1 = 5\]Therefore, the product of the eigenvalues is 5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Trace
In the world of linear algebra, the matrix trace plays a crucial role, particularly when dealing with eigenvalues. The trace of a matrix is defined as the sum of its diagonal elements. It is a simple yet powerful concept. For example, consider the matrix \(A = \begin{bmatrix} 2 & 2 & 1 \ 1 & 3 & 1 \ 1 & 2 & 2 \end{bmatrix}\). The diagonal elements are 2, 3, and 2.
The trace of this matrix is calculated by adding these diagonal elements together:- First diagonal element: 2- Second diagonal element: 3- Third diagonal element: 2
By summing these up, we get a trace of 7, i.e., \(2 + 3 + 2 = 7\). This trace is significant because, in a matrix, the sum of the eigenvalues always equals the trace. Therefore, the trace provides a quick check on the sum of eigenvalues.
Understanding this property will help you connect the dots between matrix operations and their implications in solving linear algebra problems.
The trace of this matrix is calculated by adding these diagonal elements together:- First diagonal element: 2- Second diagonal element: 3- Third diagonal element: 2
By summing these up, we get a trace of 7, i.e., \(2 + 3 + 2 = 7\). This trace is significant because, in a matrix, the sum of the eigenvalues always equals the trace. Therefore, the trace provides a quick check on the sum of eigenvalues.
Understanding this property will help you connect the dots between matrix operations and their implications in solving linear algebra problems.
Determinant Calculation
The determinant is another fundamental concept in linear algebra, providing insights into a matrix’s properties, such as whether it is invertible or the volume of the transformation it represents. It is particularly important when calculating the product of eigenvalues, as these products equal the determinant of the matrix.
Let's delve into the calculation of the determinant for the matrix \(A = \begin{bmatrix} 2 & 2 & 1 \ 1 & 3 & 1 \ 1 & 2 & 2 \end{bmatrix}\). The determinant of a \(3 \times 3\) matrix can be found using cofactor expansion. For our matrix, it can be expanded as:
- Choose a row or column, typically the first row for simplicity.- Multiply each element by the determinant of the minor matrix that remains after removing the row and column.- Alternating signs are applied thereafter: positive, negative, positive.
Thus, the calculation becomes:\[ \det(A) = 2((3 \times 2) - (1 \times 2)) - 2((1 \times 2) - (1 \times 1)) + 1((1 \times 2) - (1 \times 3)) \]Calculating each component, we find:- \(2(6 - 2)\)- \(-2(2 - 1)\)- \(+1(2 - 3)\)
Combining these, we get \(2 \cdot 4 - 2 \cdot 1 + 1 \cdot (-1) = 8 - 2 - 1 = 5\). Hence, the determinant indicates the product of the eigenvalues as 5. This understanding can save time and avoid errors in computations involving eigenvalues.
Let's delve into the calculation of the determinant for the matrix \(A = \begin{bmatrix} 2 & 2 & 1 \ 1 & 3 & 1 \ 1 & 2 & 2 \end{bmatrix}\). The determinant of a \(3 \times 3\) matrix can be found using cofactor expansion. For our matrix, it can be expanded as:
- Choose a row or column, typically the first row for simplicity.- Multiply each element by the determinant of the minor matrix that remains after removing the row and column.- Alternating signs are applied thereafter: positive, negative, positive.
Thus, the calculation becomes:\[ \det(A) = 2((3 \times 2) - (1 \times 2)) - 2((1 \times 2) - (1 \times 1)) + 1((1 \times 2) - (1 \times 3)) \]Calculating each component, we find:- \(2(6 - 2)\)- \(-2(2 - 1)\)- \(+1(2 - 3)\)
Combining these, we get \(2 \cdot 4 - 2 \cdot 1 + 1 \cdot (-1) = 8 - 2 - 1 = 5\). Hence, the determinant indicates the product of the eigenvalues as 5. This understanding can save time and avoid errors in computations involving eigenvalues.
Linear Algebra
Linear algebra is the backbone of various mathematical and practical applications, particularly when it comes to matrices and eigenvalues. It enables the quantification and solution of complex mathematical problems through vector spaces and linear mappings.
Matrices offer a way to represent systems of linear equations, transformations, and data compression among others. Their properties, like eigenvalues and eigenvectors, tell us how systems respond to external inputs or how they transform spaces.
In linear algebra, eigenvalues are pivotal, as they can reveal the stability of a system, its mode shapes in mechanical applications, or even the growth rate in population models. The eigenvalues of a matrix are special numbers associated with the matrix that indicate scaling factors along certain directions in the space.
By using the key concepts such as the matrix trace and determinant, you can efficiently determine the sum and product of eigenvalues — essential for quickly assessing an entire system's characteristics without solving complex secular equations. Thus, mastering these concepts in linear algebra can significantly empower problem-solving capabilities in various scientific and engineering fields.
Matrices offer a way to represent systems of linear equations, transformations, and data compression among others. Their properties, like eigenvalues and eigenvectors, tell us how systems respond to external inputs or how they transform spaces.
In linear algebra, eigenvalues are pivotal, as they can reveal the stability of a system, its mode shapes in mechanical applications, or even the growth rate in population models. The eigenvalues of a matrix are special numbers associated with the matrix that indicate scaling factors along certain directions in the space.
By using the key concepts such as the matrix trace and determinant, you can efficiently determine the sum and product of eigenvalues — essential for quickly assessing an entire system's characteristics without solving complex secular equations. Thus, mastering these concepts in linear algebra can significantly empower problem-solving capabilities in various scientific and engineering fields.
