Questions: From the following, select all statements that are true. There may be more than one correct answer, A, X, and D are n x n matrices, A. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. B. If there exists a basis for R^n consisting entirely of eigenvectors of A, then A is diagonalizable, C. If A is diagonalizable, then A is invertible, D. A is diagonalizable if A = X D X^-1 for some diagonal matrix D and some invertible matrix X, E. None of the above.

From the following, select all statements that are true. There may be more than one correct answer, A, X, and D are n x n matrices,
A. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
B. If there exists a basis for R^n consisting entirely of eigenvectors of A, then A is diagonalizable,
C. If A is diagonalizable, then A is invertible,
D. A is diagonalizable if A = X D X^-1 for some diagonal matrix D and some invertible matrix X,
E. None of the above.

Transcript text: From the following, select all statements that are true. There may be more than one correct answer, $A, X$, and $D$ are $n \times n$ matrices, A. $\boldsymbol{A}$ is diagonalizable if and only if $A$ has $n$ eigenvalues, counting multiplicities. B. If there exists a basis for $\mathbb{R}^{n}$ consisting entirely of eigenvectors of $A$, then $A$ is diagonalizable, C. If $A$ is diagonalizable, then $A$ is invertible, D. $A$ is diagonalizable if $A=X D X^{-1}$ for some diagonal matrix $D$ and some invertible matrix $X$, E. None of the above,

Solution

Solution Steps

To determine which statements are true, we need to analyze each statement based on the properties of diagonalizable matrices.

Statement A: A matrix $ A $ is diagonalizable if and only if it has $ n $ linearly independent eigenvectors. This is equivalent to saying $ A $ has $ n $ eigenvalues, counting multiplicities.
Statement B: If there exists a basis for $ \mathbb{R}^n $ consisting entirely of eigenvectors of $ A $, then $ A $ is diagonalizable. This is true by definition.
Statement C: If $ A $ is diagonalizable, it does not necessarily mean $ A $ is invertible. A diagonalizable matrix can have zero as an eigenvalue, making it non-invertible.
Statement D: If $ A $ is diagonalizable, then there exists an invertible matrix $ X $ and a diagonal matrix $ D $ such that $ A = X D X^{-1} $. This is true by definition.

Solution Approach

Analyze each statement based on the properties of diagonalizable matrices.
Verify the conditions for diagonalizability and invertibility.

Step 1: Analyze Statement A

A matrix $ A $ is diagonalizable if and only if it has $ n $ linearly independent eigenvectors. This is equivalent to saying $ A $ has $ n $ eigenvalues, counting multiplicities. Therefore, statement A is true.

Step 2: Analyze Statement B

If there exists a basis for $ \mathbb{R}^n $ consisting entirely of eigenvectors of $ A $, then $ A $ is diagonalizable. This is true by definition. Therefore, statement B is true.

Step 3: Analyze Statement C

If $ A $ is diagonalizable, it does not necessarily mean $ A $ is invertible. A diagonalizable matrix can have zero as an eigenvalue, making it non-invertible. Therefore, statement C is false.

Step 4: Analyze Statement D

If $ A $ is diagonalizable, then there exists an invertible matrix $ X $ and a diagonal matrix $ D $ such that $ A = X D X^{-1} $. This is true by definition. Therefore, statement D is true.

Final Answer

The true statements are: \[ \boxed{A, B, D} \]

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