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Chapter 6: Problem 8

Multiply the matrix and the vector to determine if the vector is an eigenvector. If so, what is the eigenvalue? $$ \left[\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -2 \end{array}\right],\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] $$

Short Answer

Expert verified
Yes, 5 is the eigenvalue.

Step by step solution

01

- Understand matrix-vector multiplication

To determine if a vector is an eigenvector of a matrix, multiply the matrix by the vector. If the result is a scalar multiple of the original vector, then it is an eigenvector, and the scalar is the eigenvalue.
02

- Write down the matrix and the vector

The matrix is \[\left[\begin{array}{rrr}5 & 0 & 0 \0 & 3 & 0 \0 & 0 & -2\end{array}\right]\]and the vector is \[\left[\begin{array}{l}1 \0 \0\end{array}\right]\].
03

- Multiply the matrix by the vector

Perform the multiplication: \[\left[\begin{array}{rrr}5 & 0 & 0 \0 & 3 & 0 \0 & 0 & -2\end{array}\right] \left[\begin{array}{l}1 \0 \0\end{array}\right] = \left[\begin{array}{l}5(1) + 0(0) + 0(0) \0(1) + 3(0) + 0(0) \0(1) + 0(0) + (-2)(0)\end{array}\right] = \left[\begin{array}{l}5 \0 \0\end{array}\right] \].
04

- Compare the result to the original vector

The resulting vector is \[\left[\begin{array}{l}5 \0 \0\end{array}\right]\]. This is a scalar multiple of the original vector \[\left[\begin{array}{l}1 \0 \0\end{array}\right]\]. The scalar multiple is 5.
05

- Conclude the eigenvalue

Since multiplying the matrix by the vector yields a vector that is 5 times the original vector, 5 is the eigenvalue.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Multiplication
Matrix multiplication is a basic yet crucial concept in linear algebra. When multiplying a matrix by a vector, the result should be a new vector. Here's the process:
  • Each element of the resulting vector is found by taking the dot product of each row of the matrix with the vector.
  • This involves multiplying corresponding elements and summing them up.
In the exercise, we multiply the matrix \(\begin{array}{rrr}5 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & -2\right)\) with the vector \(\begin{array}{l}1 \ 0 \ 0\right)\). Performing the multiplication step-by-step:
\(\begin{array}{rrr}5*1 + 0*0 + 0*0 \ 0*1 + 3*0 + 0*0 \ 0*1 + 0*0 + (-2)*0\right) = \begin{array}{l}5 \ 0 \ 0\right)\).
Thus, knowing how to multiply matrices is essential for verifying eigenvectors.
Eigenvalues
Eigenvalues are scalars associated with eigenvectors in matrix operations. To identify an eigenvector and its eigenvalue:
  • Multiply the matrix by the vector.
  • If the result is a scalar multiple of the original vector, the vector is an eigenvector, and the scalar is the eigenvalue.
In the provided exercise, we performed the multiplication: \(\begin{array}{rrr}5 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & -2\right) \begin{array}{l}1 \ 0 \ 0\right) \rightarrow \begin{array}{l}5 \ 0 \ 0\right)\).
The result \(\begin{array}{l}5 \ 0 \ 0\right)\) is indeed a scalar multiple (5) of the original vector \(\begin{array}{l}1 \ 0 \ 0\right)\), making 5 the eigenvalue.
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. Key concepts include:
  • Vector spaces
  • Linear transformations
  • Matrix operations like addition, multiplication, and the calculation of determinants
Understanding these concepts is vital for solving problems related to eigenvectors and eigenvalues. In the context of our exercise, a firm grasp of matrix multiplication and recognizing scalar multiples helps in identifying eigenvectors and calculating eigenvalues. Mastering these basics lays a strong foundation for more advanced topics in linear algebra.

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Most popular questions from this chapter

Show that if \(x_{n}\) and \(x_{n}^{\prime}\) are two solutions to the linear difference equation $$ x_{n+1}=a x_{n}+b x_{n-1} $$ then \(x_{n}+x_{n}^{\prime}\) and \(c x_{n}\) are also solutions, where \(c\) is a constant.

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{l} 5 \\ 5 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 4 \\ 1 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

Find all the eigenvalues and the corresponding eigenvectors for the following matrices. $$ \left[\begin{array}{rr} 1 & 0 \\ -1 & 2 \end{array}\right] $$

Solve using Gaussian elimination. \(x+y+z+w=5\) \(x+z+w=6\) \(y+z+w=4\) \(x+z \quad=3\)

Multiply the matrix and the vector to determine if the vector is an eigenvector. If so, what is the eigenvalue? $$ \left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right] \cdot\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$
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