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

To solve the problem of determining for which invertible matrices \( M \), the polynomial \( p_M(x) - p_{M^{-1}}(x) \) is constant, we need to examine the properties of the characteristic polynomial of a matrix and its inverse. The characteristic polynomial of a matrix \( A \), \( p_A(x) \), is given by:

\(\det(xI - A)\) 

where \( I \) is the identity matrix of the same dimension as \( A \).

For an invertible matrix \( M \), suppose the eigenvalues are \(\lambda_1, \lambda_2, \ldots, \lambda_n\). Then, the eigenvalues of \( M^{-1} \) are \(\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \ldots, \frac{1}{\lambda_n}\).

The characteristic polynomial of \( M \) is:

\(p_M(x) = (x - \lambda_1)(x - \lambda_2) \cdots (x - \lambda_n)\)

And the characteristic polynomial of \( M^{-1} \) is:

\(p_{M^{-1}}(x) = (x - \frac{1}{\lambda_1})(x - \frac{1}{\lambda_2}) \cdots (x - \frac{1}{\lambda_n})\)

We need \( p_M(x) - p_{M^{-1}}(x) \) to be a constant. For this to happen, under normal circumstances, the eigenvalues of \( M \) and \( M^{-1} \) must combine such that their differences are constant for all values of \( x \).

1. Check the following matrices:
   • \( M = \begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix} \)
   • \( M = \begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix} \)
   • \( M = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} \)
   • \( M = \begin{bmatrix} 5 & -8 \\ 2 & -3 \end{bmatrix} \)
2. Compute the determinant of each matrix to verify they are invertible.
3. Find the eigenvalues of each matrix using the determinant and trace. For a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the characteristic polynomial is:
4. Calculate \( p_M(x) - p_{M^{-1}}(x) \) to check if it is constant for the matrices provided.

Upon evaluation, the matrices that satisfy the condition are:

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

In these cases, the properties of the eigenvalues and their reciprocals allow the polynomial difference to remain constant (typically zero for similar matrices' inverse reciprocal pairs).