91Ó°ÊÓ

Find the particular solution. \(x_{n+1}=x_{n}+2 x_{n-1} ; x_{0}=2, x_{1}=1\)

Let \(A=\left[\begin{array}{ll}4 & -1 \\ 7 & -9\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 2 & -2\end{array}\right], C=\left[\begin{array}{rr}8 & 3 \\ 0 & -3\end{array}\right]\) \(D=\left[\begin{array}{rrr}5 & -6 & 1 \\ 10 & 3 & -1\end{array}\right], E=\left[\begin{array}{rr}-7 & 4 \\ -3 & 2 \\ 2 & -1\end{array}\right]\) \(F=\left[\begin{array}{rrr}-4 & 2 & 3 \\ 0 & -1 & 2 \\ -7 & -2 & 5\end{array}\right]\), and \(v=\left[\begin{array}{r}2 \\\ -3\end{array}\right]\). Compute \(\mathbf{A}^{3} \mathbf{v}\).

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$

Compute the inverse matrix. $$ \left[\begin{array}{rrr} 0 & -1 & -3 \\ 1 & 5 & 16 \\ 1 & 2 & 8 \end{array}\right] $$

Let \(A=\left[\begin{array}{ll}4 & -1 \\ 7 & -9\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 2 & -2\end{array}\right], C=\left[\begin{array}{rr}8 & 3 \\ 0 & -3\end{array}\right]\) \(D=\left[\begin{array}{rrr}5 & -6 & 1 \\ 10 & 3 & -1\end{array}\right], E=\left[\begin{array}{rr}-7 & 4 \\ -3 & 2 \\ 2 & -1\end{array}\right]\) \(F=\left[\begin{array}{rrr}-4 & 2 & 3 \\ 0 & -1 & 2 \\ -7 & -2 & 5\end{array}\right]\), and \(v=\left[\begin{array}{r}2 \\\ -3\end{array}\right]\). Compute \(\mathrm{B}^{2} \mathrm{v}\).

Find the particular solution. \(x_{n+1}=2 x_{n}+8 x_{n-1} ; x_{0}=1, x_{1}=4\)

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{l} 6 \\ 4 \end{array}\right], \mathbf{w}=\left[\begin{array}{r} -2 \\ 0 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 0 \\ 3 \end{array}\right] $$

Find the particular solution. \(x_{n+1}=\frac{5}{2} x_{n}-x_{n-1} ; x_{0}=0, x_{1}=-3 / 2\)

Compute the inverse matrix. $$ \left[\begin{array}{rrr} -5 & -7 & -5 \\ -1 & 0 & -2 \\ 9 & 10 & 11 \end{array}\right] $$

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ v=\left[\begin{array}{l} 3 \\ 5 \end{array}\right], w=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], u=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] $$