We know that $A = I_2 - MM^T$ and $M^TM = I_1$, which implies that $M$ is a unit vector. The matrix $A$ is a projection matrix, and for a projection matrix, the eigenvalues are either 0 or 1.
In this case, the eigenvalue $\lambda$ of $A$ can be either 0 or 1. Therefore, the sum of squares of all possible values of $\lambda$ is:
$0^2 + 1^2 = 1 + 1 = 2.$
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32