To determine if an n×n matrix A is diagonalizable, we need to check if A has n linearly independent eigenvectors.

Step 1: Find the eigenvalues of matrix A by solving the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix of size n×n.

Step 2: For each eigenvalue λ, find the corresponding eigenvectors by solving the equation (A - λI)x = 0, where x is the eigenvector.

Step 3: Count the number of linearly independent eigenvectors obtained from step 2. If the count is equal to n, then the matrix A is diagonalizable. Otherwise, it is not diagonalizable.

So, the statement "An n×n matrix A is diagonalizable if A has n linearly independent eigenvectors" is true.

Question

To determine if an n×n matrix A is diagonalizable, we need to check if A has n linearly independent eigenvectors.

Step 1: Find the eigenvalues of matrix A by solving the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix of size n×n.

Step 2: For each eigenvalue λ, find the corresponding eigenvectors by solving the equation (A - λI)x = 0, where x is the eigenvector.

Step 3: Count the number of linearly independent eigenvectors obtained from step 2. If the count is equal to n, then the matrix A is diagonalizable. Otherwise, it is not diagonalizable.

So, the statement "An n×n matrix A is diagonalizable if A has n linearly independent eigenvectors" is true.

Knowee AI · Accepted Answer

To determine if an n×n matrix A is diagonalizable, we need to check if A has n linearly independent eigenvectors.

Step 1: Find the eigenvalues of matrix A by solving the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix of size n×n.

Step 2: For each eigenvalue λ, find the corresponding eigenvectors by solving the equation (A - λI)x = 0, where x is the eigenvector.

Step 3: Count the number of linearly independent eigenvectors obtained from step 2. If the count is equal to n, then the matrix A is diagonalizable. Otherwise, it is not diagonalizable.

So, the statement "An n×n matrix A is diagonalizable if A has n linearly independent eigenvectors" is true.

An n×n matrix A is diagonalizable if A has n linearly independent eigenvectors.A. TrueB. False

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An n×n matrix A is diagonalizable if A has n linearly independent eigenvectors.

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