91Ó°ÊÓ

Find the standard matrix of the composite transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\) ? Counterclockwise rotation through \(60^{\circ}\), followed by reflection in the line \(y=x\)

Find the standard matrix of the composite transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\) ? Reflection in the \(y\) -axis, followed by clockwise rotation through \(30^{\circ}\)

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \end{array}\right]$$

Find the inverse of the given elementary matrix. $$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & 0 \\ 0 & 0 & 1 \end{array}\right], c \neq 0$$

Which of the following matrices are skew-symmetric? (a) \(\left[\begin{array}{rr}1 & 2 \\ -2 & 3\end{array}\right]\) (b) \(\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]\) (c) \(\left[\begin{array}{rrr}0 & 3 & -1 \\ -3 & 0 & 2 \\ 1 & -2 & 0\end{array}\right]\) (d) \(\left[\begin{array}{rrr}0 & 1 & 2 \\ -1 & 0 & 5 \\ 2 & 5 & 0\end{array}\right]\)

Let \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] .\) Find a formula for \(A^{n}(n \geq 1)\) and verify your formula using mathematical induction.

Find the inverse of the given elementary matrix. $$\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array}\right], c \neq 0$$

Give a componentwise definition of a skew-symmetric matrix.

Find the standard matrix of the composite transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\) ? Clockwise rotation through \(45^{\circ}\), followed by projection onto the \(y\) -axis, followed by clockwise rotation through \(45^{\circ}\)

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$$