Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where . What is the characteristic polynomial of T? 

Concept:

Linear Transformation and Matrix Representation:

T is a linear transformation that maps any  matrix  to , where  is a given  matrix.

Characteristic Polynomial:

The characteristic polynomial of a matrix A is given by the determinant of , where  is

the eigenvalue and  is the identity matrix. The characteristic polynomial is  where  is

the identity matrix of the same dimension as A, and  represents the eigenvalues.

Explanation:
A =
 

Here, T  is acting   matrices. So, we need to understand how  A acts on any , where B =  .

The action of A  on B is

T(B) = AB =  = 

This shows how the transformation T  scales the first row of the matrix B by 2 and leaves the second row unchanged.

Now, we represent T as a matrix that acts on the vectorization of the   matrix B. If we write the entries

of B as a vector  (by stacking the columns of  B), i.e.,

Then the action of T on   can be represented as a  matrix. The effect of  T  is:

This can be written as the matrix multiplication

Thus, the matrix representation of  T is

The characteristic polynomial of a matrix  T is given by:

where   is the  identity matrix. Substituting  T into this expression:

Now, we compute the determinant

Simplifying,

Thus, the characteristic polynomial of  T is

Hence the correct option is 3).