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Consider an $n \times n$ matrix $A$ with the property that the column sums all equal the same number $s$ . Show that $s$ is an eigenvalue of $A .[\text { Hint: Use Exercises } 27 \text { and } 29 .]$

Since $A$ has the property that the column sums all equal $s,$ then $A^{T}$ has property that all row sums all equal $s$ .From exercise 29 we know that $s$ is eigenvalue for $A^{T}$ .From exercise 27 we know that $A$ and $A^{T}$ have same eigenvalues, so $s$ must also be eigenvalue for $A .$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

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Harvey Mudd College

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Hello, one. So in this given question, there is when metrics A which has the property that the column sums are all equal to us. Since we all know if a has the property, that the column sums are all equal toe all equal to us. So the transports off, we'll have a property that goes some sorry equals to us all the blossoms r equals to us. Okay, So from the previous question, we can understand that a transpose will have an Eigen value off s got it. And from the other one, we know that a and eight, I suppose, has the same Eigen value. So if since eight transports is having Diagon Value s, we can say that metrics, they will also have I can value s X. Okay. As a metrics will have the same Eigen value, which is it was to us

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