Linear Algebra

Determine if a given matrix is diagonalizable and, if so, diagonalize it.

Raise the matrix $A = \begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix}$ to the 100th power using diagonalization.

Diagonalize the matrix A where A is given as $\begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{bmatrix}$.

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 $A - \lambda I = 0$.

Diagonalize the following 3x3 matrix.

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.

Determine if a given matrix is an elementary matrix by performing one row operation on an identity matrix to see if it transforms into the given matrix.

Determine if the given matrix is an Elementary matrix by checking if it can be obtained from the identity matrix using a single Elementary row operation.

Check if a set of vectors in $\mathbb{R}^3$ consisting of (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for $\mathbb{R}^3$.

Determine if the given set of four matrices, with specific ones and zeros, span $R^{2x2}$ and form a basis.

$\begin{bmatrix}1 & 1 \\0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 0 \\1 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}$ $\begin{bmatrix}0 & 1 \\1 & 1\end{bmatrix}$

Check comprehension related to vector spaces, subspaces, span, linear independence, basis, and dimension.

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 $\lambda = 3$.

Find the inverse of the given 3x3 matrix $A$ using the elementary row operation method.

Find the inverse of a 4x4 matrix using row operations.

Solve the system of linear equations using the Gauss-Jordan elimination method.

Solve the following system of equations using Gauss-Jordan elimination: \begin{align*} 2x - 5y &= 15 \\ 3x + y &= 31 \end{align*}

Using Gaussian elimination, solve the system of linear equations represented by the matrix.