A square matrix is magic if the sum of the numbers in each row, column, and diagonal is the same.
a. Find an ordered basis $B$ for the vector space $V$ of $3\times 3$ magic squares.
$B = ($ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $)$.
b. Let $L :V\to V$ be the linear transformation defined by $L(M)=M^T$. Find the matrix of $L$ in the basis $B$.
$[L]_B^B =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$
c. Find the determinant
$\det([L]_B^B) =$
d. Find the characteristic polynomial $f(x)$ of $L$.
$f(x) =$
e. Find a diagonal matrix $D$ and an invertible matrix $P$ such that $D = P^{-1} [L]_B^B P$.
$D =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $P =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$