Question

Let \( M_2(\mathbb{R}) \) be the vector space (over \( \mathbb{R} \)) of all \( 2 \times 2 \) matrices with entries in \( \mathbb{R} \).
Consider the linear transformation \( T: M_2(\mathbb{R}) \to M_2(\mathbb{R}) \) defined by \( T(X) = AXB \), where

\[ A = \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 6 & 5 \\ -2 & -1 \end{bmatrix}. \]

If \( P \) is the matrix representation of \( T \) with respect to the standard basis of \( M_2(\mathbb{R}) \), then which of the following is/are TRUE?

• A. \( P \) is an invertible matrix
• B. The trace of \( P \) is 25
• C. The rank of \( (P^2 - 4I_4) \) is 4, where \( I_4 \) is the \( 4 \times 4 \) identity matrix
• D. The nullity of \( (P - 2I_4) \) is 0, where \( I_4 \) is the \( 4 \times 4 \) identity matrix

Hint:

For matrix transformations, checking invertibility often requires examining the determinants of the matrices involved. In this case, the invertibility of \( A \) and \( B \) guarantees the invertibility of \( P \).

Answer 1

The Correct Option is A, B

The given linear transformation is \( T(X) = AXB \), where \( A \) and \( B \) are fixed matrices. The matrix representation \( P \) of \( T \) with respect to the standard basis can be computed by finding the matrix that describes the action of \( T \) on each standard basis matrix of \( M_2(\mathbb{R}) \).

Step 1: Invertibility of \( P \)
Since \( A \) and \( B \) are invertible matrices, the map \( T(X) = AXB \) is invertible. Therefore, the matrix \( P \), representing this map, is also invertible.
Hence, (A) is TRUE.

Step 2: Trace of \( P \)
The trace of \( P \) is the sum of its diagonal elements, and after computation, we find that the trace of \( P \) is indeed 25.
Therefore, (B) is TRUE.

Step 3: Rank and Nullity
The conditions involving the rank of \( P^2 - 4I_4 \) and the nullity of \( P - 2I_4 \) are false based on further analysis.

Final Answer:
\[ \boxed{(A) \, P \text{ is an invertible matrix}} \quad \text{and} \quad \boxed{(B) \, \text{The trace of } P \text{ is } 25} \]