Question

Confirm by multiplication that $x$ is an eigenvector of $A,$ and find the corresponding eigenvalue. $$A=\left[\begin{array}{ll}5 & -1 \\1 & 3\end{array}\right] ; \mathbf{x}=\left[\begin{array}{l} 1 \\1\end{array}\right]$$

Name: Problem 2, Chapter 5: Elementary Linear Algebra: Applications Version
Uploaded: 2021-05-05T23:50:01-08:00
Duration: 2 min 38 s
Channel: Andrija Isakov
Description: Confirm by multiplication that x is an eigenvector of A, and find the corresponding eigenvalue. A=[ 5 -1 1 3 ] ; 𝐱=[ 1 1 ]

   Confirm by multiplication that $x$ is an eigenvector of $A,$ and find the corresponding eigenvalue.
$$A=\left[\begin{array}{ll}5 & -1 \\1 & 3\end{array}\right] ; \mathbf{x}=\left[\begin{array}{l}
1 \\1\end{array}\right]$$

Elementary Linear Algebra: Applications Version

Howard Anton, Chris… 11th Edition

Chapter 5, Problem 2

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Solved by Expert Andrija Isakov

05/05/2021

Step 1

$$A\mathbf{x}=\left[\begin{array}{ll}5 & -1 \\1 & 3\end{array}\right] \left[\begin{array}{l}1 \\1\end{array}\right]$$ Show more…

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Confirm by multiplication that $x$ is an eigenvector of $A,$ and find the corresponding eigenvalue. $$A=\left[\begin{array}{ll}5 & -1 \\1 & 3\end{array}\right] ; \mathbf{x}=\left[\begin{array}{l} 1 \\1\end{array}\right]$$

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Key Concepts

Eigenvalue

An eigenvalue of a matrix is a scalar that indicates how much its corresponding eigenvector is scaled during the transformation represented by the matrix. When a matrix multiplies an eigenvector, the output is the eigenvector multiplied by its eigenvalue, satisfying the equation Ax = ?x.

Eigenvector

An eigenvector is a nonzero vector that, when a matrix transforms it through multiplication, only gets scaled by a constant factor (the eigenvalue) without any change in direction. This property makes eigenvectors critical in understanding the behavior of linear transformations.

Matrix Multiplication Verification

Matrix multiplication verification is the process of multiplying a matrix by a candidate eigenvector to check if the resulting vector is a scalar multiple of the original. This confirms the relationship Ax = ?x, where the scalar factor ? is the eigenvalue corresponding to the eigenvector x.

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