​Let V be the real vector space of 2 × 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?

Correct option is C

Solution:

$$\textbf{The action of A on B: } T(B) = AB = \begin{pmatrix}2 & 0 \\0 & 1\end{pmatrix}\begin{pmatrix}b_{11} & b_{12} \\b_{21} & b_{22}\end{pmatrix}\\=\begin{pmatrix}2b_{11} & 2b_{12} \\b_{21} & b_{22}\end{pmatrix}\\\text{This shows how the transformation T scales the first row of the matrix B by 2 and leaves the second row unchanged.}\\\textbf{Representing T as a matrix acting on the vectorization of B: }\text{Let } \mathbf{b} \in \mathbb{R}^4 \text{ by stacking the columns of B, i.e.,}\mathbf{b} = \begin{pmatrix}b_{11} \\b_{21} \\b_{12} \\b_{22}\end{pmatrix}\\\text{Then, the action of T on } \mathbf{b} \text{ can be represented as:}T(\mathbf{b}) = \begin{pmatrix}2b_{11} \\b_{21} \\2b_{12} \\b_{22}\end{pmatrix}\\\ \\\text{This can also be written as a matrix multiplication:}T(\mathbf{b}) =\begin{pmatrix}2 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 2 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}b_{11} \\b_{21} \\b_{12} \\b_{22}\end{pmatrix}$$

Thus, the matrix representation of  T is

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