List of practice Questions

For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:

\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \).
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let

\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).

For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form: \[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \). Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let \[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).