Linear Algebra: Eigenvalues and Eigenvectors

Consider the matrix $A$, with entries $-3, -4; 5, 6$. We must first find the eigenvalues, which means we must solve for the values of $\lambda$ that satisfy this expression, where the determinant of $A - \lambda I = 0$.

For a given matrix $A$, find the eigenvalues $\lambda$ and corresponding eigenvectors $X$ such that $A X = \lambda X$.

Find the eigenvalues and eigenvectors of the given matrix.

Assume we have five websites numbered 1, 2, 3, 4, and 5. Determine which website will have more traffic using the PageRank algorithm. Construct the Google Matrix for these websites and analyze it to find the relative importance or ranking of each website based on their traffic. Use eigenvalues and eigenvectors for the analysis.

Using a 2x2 matrix, identify eigenvectors and eigenvalues. Explain the transformation effects on vectors when multiplied with this matrix, including stretching, shrinking, and rotation dynamics. Consider cases with real and non-real eigenvalues and describe their implications for vector transformations.

Given this matrix $A$, find the eigenvector corresponding to the eigenvalue $\lambda = 3$.