Question

Suppose p(λ) = λ(λ−1)(λ+2)(λ+1) is the characteristic polynomial of a matrix A. Consider the statements:

Answer 1

Answer:  The false statement is

III. A is invertible.

Step-by-step explanation:  Given that the characteristic polynomial of a matrix A is

$p(\lambda)=\lambda(\lambda-1)(\lambda+2)(\lambda+1).$

We are to select the statement that is FALSE.

The characteristic values of A are given by

$p(\lambda)=0\\\\\Rightarrow \lambda(\lambda-1)(\lambda+2)(\lambda+1)=0\\\\\Rightarrow \lambda=0,1,-2,-1.$.

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 $A=P^{-1}DP,$ 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,

$|A|=0\times1\times(-2)\times(-1)=0.$

This implies that A is non-invertible.

So, statement III is FALSE.

Thus, statement III is FALSE.