Question

Let \( p_A(x) \) denote the characteristic polynomial of a square matrix \( A \). Then, for which of the following invertible matrices \( M \), the polynomial \( p_M(x) - p_{M^{-1}}(x) \) is constant?

• A. \( M = \begin{bmatrix} 5 & 7 2 & 3 \end{bmatrix} \)
• B. \( M = \begin{bmatrix} 3 & 1 4 & 2 \end{bmatrix} \)
• C. \( M = \begin{bmatrix} 1 & 2 3 & -1 \end{bmatrix} \)
• D. \( M = \begin{bmatrix} 5 & -8 2 & -3 \end{bmatrix} \)

Hint:

The characteristic polynomial of a matrix \( M \) and its inverse \( M^{-1} \) are related through \( p_{M^{-1}}(x) = x^n p_M\left( \frac{1}{x} \right) \), which allows you to determine when their difference is constant.

Answer 1

The Correct Option is A, C, D

The characteristic polynomial of a square matrix \( A \), denoted by \( p_A(x) \), is given by:

\[ p_A(x) = \det(xI - A) \]
We are tasked with finding for which matrices \( M \) the polynomial \( p_M(x) - p_{M^{-1}}(x) \) is constant. The key property is that for the inverse matrix \( M^{-1} \), the characteristic polynomial is related to that of \( M \) by the following relationship:

\[ p_{M^{-1}}(x) = x^n p_M\left( \frac{1}{x} \right) \]
where \( n \) is the size of the matrix \( M \). The polynomial \( p_M(x) - p_{M^{-1}}(x) \) will be constant if the matrices satisfy the conditions derived from this relationship.

Upon evaluating the characteristic polynomials for the given matrices, we find that the condition holds for matrices:

\( M = \begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix} \)
\( M = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} \)
\( M = \begin{bmatrix} 5 & -8 \\ 2 & -3 \end{bmatrix} \)

Thus, the correct answer is:

\[ \boxed{ \text{(A)} \, M = \begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix}, \quad \text{(C)} \, M = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}, \quad \text{(D)} \, M = \begin{bmatrix} 5 & -8 \\ 2 & -3 \end{bmatrix} } \]