Linear Algebra
Algebra
Level
2
If the characteristic polynomial of a matrix is , then the matrix is not diagonalizable.
False
True
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1 solution
The eigenvalues can be seen in the roots of the polynomial. It is known that a matrix whose eigenvalues are all unique is diagonalizable. However, the inverse is not true in general - if a matrix has an eigenvalue with higher multiplicity than one, nothing can be said about whether it is diagonalizable. Therefore, the statement is false - the characteristic polynomial (and the eigenvalues) says nothing about whether it is diagonalizable.
Compare this with any identity matrix - it only has one distinct eigenvalue, but is nonetheless diagonalizable.