Question

Let \[ M = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix} \] Which of the following are TRUE?

• A. \( M \) is singular
• B. \( M^{-1} = \frac{1}{4} M^2 - \frac{3}{2} M + \frac{9}{4} I \), where \( I \) is the identity matrix of order 3
• C. \( M \) has three distinct eigenvalues
• D. \( M \) has three linearly independent eigenvectors

Hint:

A matrix is singular if its determinant is zero. To find the inverse, verify that the equation involving \( M^{-1} \) holds true.

Answer 1

The Correct Option is B, D

Step 1: Checking if \( M \) is singular.
A matrix is singular if its determinant is zero. The determinant of \( M \) is calculated as follows: \[ \text{det}(M) = 2 \times \left| \begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right| - (-1) \times \left| \begin{matrix} -1 & -1 \\ 1 & 2 \end{matrix} \right| + 1 \times \left| \begin{matrix} -1 & 2 \\ 1 & -1 \end{matrix} \right| \] After solving, you will find that \(\text{det}(M) \neq 0\), so \(M\) is not singular, making option (A) incorrect.
Step 2: Verifying the given equation for \( M^{-1} \).
We can verify that the given equation holds true for the inverse of \( M \), and after calculation, it turns out to be correct. Therefore, option (B) is true.
Step 3: Checking the eigenvalues of \( M \).
The characteristic equation for \( M \) is derived by solving \( \text{det}(M - \lambda I) = 0 \), where \( \lambda \) is the eigenvalue. After solving, you will find that \( M \) has repeated eigenvalues, so option (C) is false.
Step 4: Checking the eigenvectors.
Since \( M \) has three distinct eigenvalues, it will have three linearly independent eigenvectors. Therefore, option (D) is true.
Step 5: Conclusion.
Thus, the correct answers are (B) and (D).