Respuesta :
Answer: The false statement is
III. A is invertible.
Step-by-step explanation: Given that the characteristic polynomial of a matrix A is
[tex]p(\lambda)=\lambda(\lambda-1)(\lambda+2)(\lambda+1).[/tex]
We are to select the statement that is FALSE.
The characteristic values of A are given by
[tex]p(\lambda)=0\\\\\Rightarrow \lambda(\lambda-1)(\lambda+2)(\lambda+1)=0\\\\\Rightarrow \lambda=0,1,-2,-1.[/tex].
I. The characteristic polynomial of A has degree 4, so the order of A is 4. That is, the matrix A must be a 4 × 4 matrix.
So, the statement I is true.
II. Since all the characteristic values of A are all distinct, so there will be 4 linearly independent characteristic vectors.
Therefore, there exists an invertible matrix P such that [tex]A=P^{-1}DP,[/tex] where D is daigonal with characteristic values as diagonal matrix.
This makes A diagonalizable.
So, the statement II is TRUE.
III. We know that the determinant of a matrix is equal to the product of the characteristic values of the matrix.
Therefore,
[tex]|A|=0\times1\times(-2)\times(-1)=0.[/tex]
This implies that A is non-invertible.
So, statement III is FALSE.
Thus, statement III is FALSE.
