Question

Let \[ M = \begin{pmatrix} 3 & -2 & 0 \\ 2 & 3 & 3 \\ 4 & -1 & x \end{pmatrix} \] for some real number \( x \). Suppose that \( -2 \) and \( 3 \) are eigenvalues of \( M \).
If \[ M^3 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 125 \\ 125 \end{pmatrix}, \] then which one of the following is TRUE?

• A. \( x = 5 \), and the matrix \( M^2 + M \) is invertible
• B. \( x \neq 5 \), and the matrix \( M^2 + M \) is invertible
• C. \( x = 5 \), and the matrix \( M^2 + M \) is NOT invertible
• D. \( x \neq 5 \), and the matrix \( M^2 + M \) is NOT invertible

Hint:

Check the determinant of a matrix to determine if it is invertible. If the determinant is zero, the matrix is not invertible.

Answer 1

The Correct Option is B

Step 1: Use the information about eigenvalues.
The matrix \( M \) has eigenvalues \( -2 \) and \( 3 \).
We also know that \[ M^3 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 125 \\ 125 \end{pmatrix}. \] Step 2: Solve for \( x \).
The information given suggests that \( x = 5 \).
Using this value, we find that the matrix \( M^2 + M \) is not invertible because its determinant is 0.

Final Answer:
\[ \boxed{x = 5, \text{ and the matrix } M^2 + M \text{ is NOT invertible}} \]