91Ó°ÊÓ

Compute the inverse matrix. $$ \left[\begin{array}{lll} 1 & -1 & -1 \\ 5 & -1 & -7 \\ 4 & -2 & -6 \end{array}\right] $$

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{r} 8 \\ 10 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 0 \\ 1 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right] $$

Solve using Gaussian elimination. $$ \begin{array}{r} 7 x-y-9 z=1 \\ 2 x-4 z=-4 \\ -4 x+6 z=-3 \end{array} $$

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

Compute the inverse matrix. $$ \left[\begin{array}{rrr} -4.5 & -4.5 & -4 \\ 8.5 & 7.5 & 8 \\ -2 & -1 & -2 \end{array}\right] $$

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 (BD) \(\mathbf{F}\) and \(\mathrm{B}(\mathrm{DF})\) to verify the associative property for these matrices.

Find all the eigenvalues and the corresponding eigenvectors for the following matrices. $$ \left[\begin{array}{rr} 1 & 0 \\ -1 & 2 \end{array}\right] $$

Find all the eigenvalues and the corresponding eigenvectors for the following matrices. $$ \left[\begin{array}{ll} 2 & 0 \\ 1 & 3 \end{array}\right] $$

Solve using Gaussian elimination. $$ \begin{array}{r} y+3 z=2 \\ x+y+6 z=0 \\ x+2 z=4 \end{array} $$

Find the particular solution. \(x_{n+1}=5 x_{n}+6 x_{n-1} ; x_{0}=17, x_{1}=32\)