If matrix $$A$$ is positive definite, then is matrix $$A^2$$ also positive definite?

For a matrix to be positive definite, it has to be symmetric and must have all positive eigenvalues. A positive definite matrix is diagonalizable as such, $$A= Q\Lambda Q^T$$ where $$Q$$ is an orthogonal matrix and $$\Lambda$$ is the diagonal matrix of eigenvalues of $$A$$. Using diagonalization, $$A^2 = Q\Lambda^2 Q^T$$ Since every eigenvalue, $$\lambda_i$$, of $$A$$ is positive, then every $$\lambda_i^2$$ must also be positive. Therefore, $$Q\Lambda^2 Q^T$$ is a valid diagonalization of a positive definite matrix. meaning that $$A^2$$ is also positive definite where the eigenvalues of $$A^2$$ are $$\lambda_i^2$$.

Evaluate this indefinite integral. $(\int \frac{1}{1+x^2} dx$)

Using substitution: $$x= tan\theta$$ $$dx = sec^2\theta d\theta$$ $(\int \frac{1}{1+tan^2\theta}\cdot sec^2\theta d\theta$) $(\int \frac{1}{sec^2\theta}\cdot sec^2\theta d\theta$) $(\int d\theta$) $(\theta +C $) Now, since $$x=tan\theta$$ Then, $$\theta = arctanx$$ $(\int \frac{1}{1+x^2} dx = arctanx+C$)

Give an intuition as to how a change in interest rate affects the money supply in the economy.

Money supply in the economy is dependent on how many bonds are held by the citizens of the country. The income of a person can be held as either money or bonds, where bonds are issued by the government and pay back with increased value given the interest rate. If the interest rate were to rise, more people would put their income into bonds, rather than money, meaning that the money supply would decrease. Vice versa, a decrease in interest rate increases the money supply using similar logic.