91Ó°ÊÓ

Solve using Gaussian elimination. $$ \begin{array}{rr} x+y-2 z= & 4 \\ 4 x+7 y+3 z= & 3 \\ 14 x+23 y+5 z= & 10 \end{array} $$

Compute the determinant of each matrix. Determine if the matrix is invertible without computing the inverse. $$ \left[\begin{array}{lll} 1 & 1 & 1 \\ 2 & 2 & 1 \\ 3 & 3 & 1 \end{array}\right] $$

Solve using Gaussian elimination. $$ \begin{array}{r} x-y+3 z=2 \\ 2 x+3 y-z=5 \\ -x-9 y+11 z=1 \end{array} $$

Find all the eigenvalues and the corresponding eigenvectors for the following matrices. $$ \left[\begin{array}{rrr} 12 & -16 & 0 \\ 8 & -12 & 0 \\ 6 & -12 & 2 \end{array}\right] $$

Compute the determinant of each matrix. Determine if the matrix is invertible without computing the inverse. $$ \left[\begin{array}{rrr} -1 & 4 & 0 \\ 0 & 3 & -2 \\ 5 & 8 & 7 \end{array}\right] $$

Solve using Gaussian elimination. $$ \begin{array}{rr} x-y+3 z= & 2 \\ 2 x+3 y-z= & 5 \\ -x-9 y+11 z= & -4 \end{array} $$

Find all the eigenvalues and the corresponding eigenvectors for the following matrices. $$ \left[\begin{array}{rrr} 8 & 0 & 12 \\ -8 & 0 & -12 \\ -2 & 0 & -2 \end{array}\right] $$

Solve using Gaussian elimination. $$ \begin{array}{r} x-2 y-5 z=0 \\ 2 x+3 y+15 z=0 \\ -2 x-y-8 z=1 \end{array} $$

Compute the determinant of each matrix. Determine if the matrix is invertible without computing the inverse. $$ \left[\begin{array}{rrr} 0 & 4 & 2 \\ -1 & 6 & 2 \\ -1 & 14 & 6 \end{array}\right] $$

Find all the eigenvalues and the corresponding eigenvectors for the following matrices. $$ \left[\begin{array}{rrr} -3 & 8 & 6 \\ -4 & 9 & 6 \\ 3 & -6 & -4 \end{array}\right] $$