91Ó°ÊÓ

Find \(A B\) and \(B A\) if $$ \begin{gathered} A=\left[\begin{array}{ccc} 1 & i & i \\ 1+i & 1 & -i \\ i & 1+i & 1-i \end{array}\right] \text { and } \\ B=\left[\begin{array}{ccc} -1 & 1+i & i \\ 2+i & i & 1-i \\ i & 1 & i \end{array}\right] \end{gathered} $$

Find a unitary matrix U and a diagonal matrix \(D\) such that \(D=U^{-1} A U\) for the given matrix \(A\).\(A=\left[\begin{array}{cc}1 & 1+i \\ 1-i & 2\end{array}\right]\)

Find a unitary matrix U and a diagonal matrix \(D\) such that \(D=U^{-1} A U\) for the given matrix \(A\).\(A=\left[\begin{array}{cc}9 & 3-i \\ 3+i & 0\end{array}\right]\)

Find \(A^{2}\) and \(A^{4}\) if \(A=\left[\begin{array}{cc}1 & 1+i \\ -1+i & i\end{array}\right]\)

Find \(|z|\) and \(\bar{z}\), and verify that \(z \bar{z}=|z|^{2}\), if a. \(z=2+i\), b. \(z=3-4 i\).

Determine whether the given matrix is a Jordan canonical form.\(\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4\end{array}\right]\)

Find a unitary matrix U and a diagonal matrix \(D\) such that \(D=U^{-1} A U\) for the given matrix \(A\).\(A=\left[\begin{array}{rrr}0 & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\)

Show that \(z\) is a real number if and only if \(z=\bar{z}_{\text {. }}\)

Find \(A^{-1}\) if \(A=\left[\begin{array}{cc}1 & i \\ 1+i & 2+i\end{array}\right]\).

Determine whether the given matrix is a Jordan canonical form.\(\left[\begin{array}{rrrr}i & 1 & 0 & 0 \\ 0 & -i & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 3\end{array}\right]\)