Question

Question: 5. Show that if v is an eigenvector of an \(n \times n\) matrix A and v corresponds to a nonzero eigenvalue of A, then v is in Col A. (Hint: Use the definition of an eigenvector.)

Short answer

It is proved thatif v is an eigenvector of an\(n \times n\)matrix A and v corresponds to a nonzero eigenvalue of A, then v is in Col A.

Step-by-step solution

Apply Eigenvalue and Eigenvector rule

By the definition of eigenvalue and eigenvector:

\(Av = \lambda v\)

Here, \(A\) denotes the matrix,\(\lambda \) is the eigenvalue and \(v\) is the vector.

The condition for solving matrix

Multiply both sides by inverse of λ as:

\(\begin{array}{c}{\lambda ^{ - 1}}Av = {\lambda ^{ - 1}}\lambda v\\{\lambda ^{ - 1}}Av = v\\v = A\left( {{\lambda ^{ - 1}}v} \right)\end{array}\)

So, it is proved that \(v = A\left( {{\lambda ^{ - 1}}v} \right)\) is consistent, hence the vector \(v\) is linear combination of columns of matrix \(A\).