Question

Consider the 2 X 2 matrix M= \(\begin{pmatrix} 0 & a \\ a & b \end{pmatrix}\), where a,b\(\gt\)0. Then,

• A. M is a real symmetric matrix
• B. One of the eigenvalues of M is greater than b
• C. One of the eigenvalues of M is negative
• D. Product of eigenvalues of M is b

Answer 1

The Correct Option is A, B, C

To analyze the given 2 x 2 matrix \( M = \begin{pmatrix} 0 & a \\ a & b \end{pmatrix} \), with \( a, b > 0 \), we need to check each of the statements provided in the options:

1. M is a real symmetric matrix: 
   A matrix is symmetric if it is equal to its transpose. Let's compute the transpose of matrix \( M \): \(\begin{pmatrix} 0 & a \\ a & b \end{pmatrix}^T = \begin{pmatrix} 0 & a \\ a & b \end{pmatrix}\). Since the transpose of \( M \) is equal to \( M \) itself, matrix \( M \) is indeed a real symmetric matrix.
2. One of the eigenvalues of M is greater than b: 
   To find the eigenvalues of the matrix \( M \), we solve the characteristic equation: \(\det(M - \lambda I) = 0\), where \( I \) is the identity matrix.

\[\begin{vmatrix} -\lambda & a \\ a & b - \lambda \end{vmatrix} = 0\]

1. . 
   The determinant is calculated as: \(-\lambda(b - \lambda) - a^2 = \lambda^2 - b\lambda - a^2 = 0\). Solving this quadratic equation: \(\lambda = \frac{b \pm \sqrt{b^2 + 4a^2}}{2}\). The larger root: \(\lambda_1 = \frac{b + \sqrt{b^2 + 4a^2}}{2}\) is clearly greater than \( b \), since \( a, b > 0 \).
2. One of the eigenvalues of M is negative: 
   Using the solutions from before, the smaller eigenvalue is: \(\lambda_2 = \frac{b - \sqrt{b^2 + 4a^2}}{2}\). Since \( a > 0 \), the second term \(\sqrt{b^2 + 4a^2}\) is greater than \( b \), hence \(\lambda_2\) is negative.
3. Product of eigenvalues of M is b: 
   The product of the eigenvalues of a matrix is equal to its determinant. Calculating the determinant of \( M \): \(\det M = \begin{vmatrix} 0 & a \\ a & b \end{vmatrix} = 0 \times b - a \times a = -a^2\). Therefore, the product of the eigenvalues is \(-a^2\), not \( b \).

Therefore, the correct statements about the matrix \( M \) are:

• M is a real symmetric matrix.
• One of the eigenvalues of M is greater than b.
• One of the eigenvalues of M is negative.