1.
Find ranks of matrices in Exercises 1.4.
Exercises 1.6 Exercises
2.
Given \(A,B\in M_n(\K)\) show that \(\tr(A+B)=\tr(A)+\tr(B)\) and that
\begin{equation*}
\tr(AB)=\tr(BA).
\end{equation*}
Furthermore show that if \(A\) is invertible then
\begin{equation*}
\tr(B)=\tr(ABA^{-1}).
\end{equation*}
Hint.
3.
Consider any square matrix of size \(n\) of your choice and find its determinant.
4.
Show that the following matrix \(P\) is invertible, and that \(P^{-1}=P^t\text{.}\)
\begin{equation*}
P=\begin{pmatrix}0\amp 0\amp 0\amp 1\\1\amp 0\amp 0\amp 0\\0\amp 1\amp 0\amp 0\\0\amp 0\amp 1\amp 0\end{pmatrix}
\end{equation*}
5.
Show that the determinant of the following matrix is \((a-b)(b-c)(c-a)\text{.}\)
\begin{equation*}
\begin{pmatrix}1\amp 1\amp 1\\a\amp b\amp c\\a^2\amp b^2\amp c^2\end{pmatrix}
\end{equation*}
Find a condition on \(a,b,c\in\K\) so that the determinant of the above matrix is nonzero.
