Let A be an nxn matrix. Determine whether the statement below is true or false. Justify the answer. If A is invertible, then A is diagonalizable. Choose the correct answer below. O A. The statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable. O B. The statement is false. Invertible matrices always have a maximum of n linearly independent eigenvectors, making it not diagonalizable. O C. The statement is true. If a matrix is invertible, then it has n linearly independent eigenvectors, making it diagonalizable. O D. The statement is true. A diagonalizable matrix is invertible, so an invertible matrix is diagonalizable.
Let A be an nxn matrix. Determine whether the statement below is true or false. Justify the answer. If A is invertible, then A is diagonalizable. Choose the correct answer below. O A. The statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable. O B. The statement is false. Invertible matrices always have a maximum of n linearly independent eigenvectors, making it not diagonalizable. O C. The statement is true. If a matrix is invertible, then it has n linearly independent eigenvectors, making it diagonalizable. O D. The statement is true. A diagonalizable matrix is invertible, so an invertible matrix is diagonalizable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let A be an nn matrix. Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then A is diagonalizable.
Transcribed Image Text:Let A be an nxn matrix. Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then A is diagonalizable.
Choose the correct answer below.
O A. The statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable.
O B. The statement is false. Invertible matrices always have a maximum of n linearly independent eigenvectors, making it not diagonalizable.
Oc. The statement is true. If a matrix is invertible, then it has n linearly independent eigenvectors, making it diagonalizable.
O D. The statement is true. A diagonalizable matrix is invertible, so an invertible matrix is diagonalizable.
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