Two n×n matrices A and B have a common eigenvalue 2, and the same cor responding nonzero eigenvector. Which of the following options is/are correct? (Note: I is the n×n identity matrix.)

The correct options are:
1. Determinant ( A - 2I ) = 0 (represented as Determinant ( A 2I ) = 0 in the question)
2. Determinant ( B - 2I ) = 0 (represented as Determinant ( B 2I ) = 0 in the question)
3. Determinant ( A + B - 4I ) = 0 (represented as Determinant ( A + B 4I ) = 0 in the question)

Step-by-Step Explanation:

Let $v$ be the common nonzero eigenvector of the $n\times n$ matrices $A$ and $B$ corresponding to the eigenvalue $2$.
By definition of eigenvalues and eigenvectors, we have:
$$Av=2v$$
and
$$Bv=2v$$

1. Analyzing Matrix A:
From the equation $Av=2v$, we can rewrite it using the identity matrix $I$ as:
$$Av-2Iv=0$$
Factoring out the eigenvector $v$:
$$(A-2I)v=0$$
Since $v$ is a nonzero vector, the matrix $A-2I$ must be singular (non-invertible). Therefore, its determinant is equal to zero:
$$det(A-2I)=0$$
This confirms that the first option is correct.

2. Analyzing Matrix B:
Similarly, from the equation $Bv=2v$, we can write:
$$(B-2I)v=0$$
Since $v$ is a nonzero vector, the matrix $B-2I$ is also singular, which implies:
$$det(B-2I)=0$$
This confirms that the second option is correct.

3. Analyzing Matrix (A + B):
Now we consider the action of the sum of the two matrices, $A+B$, on the common eigenvector $v$:
$$(A+B)v=Av+Bv$$
Substituting $Av=2v$ and $Bv=2v$:
$$(A+B)v=2v+2v=4v$$
This shows that $4$ is an eigenvalue of the matrix $A+B$ associated with the eigenvector $v$.
We can rewrite this relation as:
$$(A+B)v-4Iv=0$$
$$(A+B-4I)v=0$$
Since $v$ is nonzero, the matrix $A+B-4I$ is singular. Therefore, its determinant is:
$$det(A+B-4I)=0$$
This confirms that the fourth option is correct, while the third option, $det(A+B-2I)=0$, is not generally correct because $2$ is not necessarily an eigenvalue of $A+B$ for the eigenvector $v$.