1 point) Are the following statements true or false? ? True False 1. AA is diago
ID: 3005089 • Letter: 1
Question
1 point) Are the following statements true or false?
? True False 1. AA is diagonalizable if A=XDX?1A=XDX?1 for some diagonal matrix DD and some invertible matrix XX.
? True False 2. AA is diagonalizable if and only if AA has nn eigenvalues, counting multiplicities.
? True False 3. If there exists a basis for RnRn consisting entirely of eigenvectors of AA, then AA is diagonalizable.
? True False 4. If AA is diagonalizable, then AA is invertible.
**********POST LOGICAL SOLUTION OR NO POINTS**********
(1 point) Are the following statements true or false? A, X and D are n x n matrices. 1 . A is diagonalizable if A XDX 1 for some diagonal matrix D and some invertible matrix X 2. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. 3. If there exists a basis for Rn consisting entirely of eigenvectors of A, then A is diagonalizable 4. If A is diagonalizable, then A is invertible. ? ? 3. If there exists a basis for R" consisting entirely of eigenvectors of A, then A is diagonalizable. ?Explanation / Answer
1. True
2. False, It always has n eigenvalues, counting multiplicity.
3. True, In this case we can construct a P which will be invertible. And a D.
4. False, It's invertible if it doesn't have zero an eigenvector but this doesn't affect diagonalizability.
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