$3\mathrm{.}2\mathrm{.}1\mathrm{.}1$ Arrays as matrices

Import numpy as $np$

Define A to be a $44$ matrix containing $1$ as every entry. Then, use A to define B$,$ a $44$ matrix containing $2$ as every entry.

Define c to be a vector containing each integer from $0$ to $15$ and reshape it into a $44$ matrix, $C.$

Use the NumPy function trace$()$ to confirm that the traces of A$,$ B and C are $4,8$ and $30$ respectively.

What does the NumPy function eye $()$ create?

NumPy has a submodule for linear algebra called linalg. Using the determinant function $)$ in this submodule, confirm that the determinants of A$,$ B and C are all zero. Comment on your results.

Define the matrix, Csq$,$ whose entries are each the square root of the corresponding entries of C$.$ Use a function from numpy. linalg to find its inverse and evaluate the product $C_{sq}*C_{sq}^{-1}.$ Comment on your results.

The function eig$()$ in numpy. linalg computes eigenvalues and eigenvectors of its argument. How are eigenvalues and eigenvectors for A and B related? Show that $A{v}_i={}_i{v}_i,$iin $\{1,2,3,4\},$ where ${}_i$ and $v_i$ are eigenvalues and eigenvectors of the matrix $A,$ respectively.