What is the shortest way to find the inverse of a 2x2 matrix?

There is an easy formula for the inverse of a 2x2 matrix. First, write down the matrix: $( A= \left[ \begin{array}{cc} a&b \\ c& d\\ \end{array} \right]. $) Then, $( A^{-1}= \left( \frac{1}{a \cdot d- c \cdot b} \right) \cdot \left[ \begin{array}{cc} d&-b \\ -c& a\\ \end{array} \right]. $) Note that the term $$\left( \frac{1}{a \cdot d- c \cdot b} \right) $$ is 1 divided by the determinant of $$A$$. Therefore, the inverse of $$A$$ exists if and only if the determinant of $$A$$ is not zero!

How do I differentiate $$y$$ with respect to $$x$$, given the equation $$y^2+x^2=1$$?

To do so, you need to use implicit differentiation. First, differentiate both sides of the equation with respect to $$x$$, thinking of $$y$$ as being a function of $$x$$. You will need to use the chain rule on the left side: $( 2\cdot y \cdot \frac{dy}{dx} =2 \cdot x. $) Now, isolate the derivative by dividing both sides by $$2 \cdot y$$: $( \frac{dy}{dx}= \frac{x}{y}.$) Note that the the value of $$ \frac{dy}{dx}$$ depends on both the value of $$x$$ AND the value of $$y$$, because $$y$$ is not actually a function of $$x$$ in the equation $$y^2+x^2=1$$.

What is the "crowding-out" effect in macroeconomics?

The "crowding-out" effect refers to the reduction in private investment after the government increases its deficit; when a government increases its deficit, it has to borrow more money. This increased demand for loanable funds increases interest rates (think of the interest rate as the "price" of a loan), so private borrowers cannot afford to borrow as much money for investing.