Exam 47: Matrices and Systems of Equations

Write the system of linear equations represented by the augmented matrix.(Use variables x,y,z,and w. ) $\left[\begin{array}{rrrrl}1 & 0 & 0 & 4 & :-5\\3 & -7 & 0 & 0 & :3\\0&-4&8&-1&:-6 \\0 & 0 & 3 & 7 &:7\end{array}\right]$

Solve the system using Gauss-Jordan elimination.​ $\left\{ \begin{array} { l } w + x = 9 \\w + y = 0 \\x + z = 0\end{array} \right.$

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.​ $\left\{ \begin{array} { r l c } - x + y - z & = - 20 \\2 x - y + z & = 29 \\3 x + 2 y + z & = 29\end{array} \right.$ ​

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}7 x - 6 y & = - 40 \\- 5 x + 7 y & = 34\end{aligned} \right.$

Solve the system by Gauss - Jordan elimination.​ $\left\{ \begin{aligned}\frac { 1 } { 3 } x + \frac { 3 } { 4 } y - \frac { 2 } { 3 } z & = - 4 \\x + \frac { 1 } { 2 } y + \frac { 1 } { 3 } z & = 30 \\\frac { 1 } { 6 } x - \frac { 1 } { 8 } y - z & = - 30\end{aligned} \right.$ ​

Determine the order of the matrix. $\left[ \begin{array} { r r r } 6 & - 7 & 7 \\9 & 6 & 7\end{array} \right]$

Perform the sequence of row operations on the matrix.What did the operations accomplish?​ $\left[ \begin{array} { c c c } 1 & 1 & - 7 \\4 & 5 & - 7 \\1 & 3 & 35\end{array} \right]$ ​ Add -4 times R₁ to R₂, Add -1 times R₁ to R₃, Add -2 times R₂ to R₃, Add -1 times R₂ to R₁. ​

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.​ $\left\{ \begin{array} { l } x + 2 y = 6 \\2 x + y = 9\end{array} \right.$ ​

Fill in the blank(s)using elementary row operations to form a row-equivalent matrix. ​​ $\left[ \begin{array} { c c c } 6 & 9 & 4 \\5 & - 4 & 7\end{array} \right]$ ​ $\left[ \begin{array} { c c c } 2 & \cdots & \frac { 4 } { 3 } \\5 & - 4 & 7\end{array} \right]$ ​

Determine whether the matrix is in row-echelon form.If it is,determine if it is also in reduced row-echelon form. $\left[ \begin{array} { r r r r } 1 & 8 & - 8 & 4 \\0 & 1 & 0 & 6 \\0 & 0 & 1 & - 9\end{array} \right]$

Select the augmented matrix for the system of linear equations.​ $\left\{ \begin{aligned}8 x - 4 y + z & = 14 \\16 x - 9 z & = 12\end{aligned} \right.$ ​

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { r r r } 4 & 5 & 11 \\3 & 1 & - 11 \\3 & 5 & 17\end{array} \right]$

Find the system of linear equations represented by the augmented matrix.Then use back substitution to solve.(Use variables x,y,z and if applicable. )​ $\left[\begin{array}{l}{\begin{array}{cccc}1 & -1 & 6 & \vdots8\end{array}} \\\begin{array}{llll}0 & 1 & -1 & \vdots6\end{array} \\\begin{array}{llll}0 & 0 & -1 & \vdots-6\end{array} \\\end{array}\right]$ ​

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.​ $\left\{ \begin{aligned}x - 4 y + 3 z - 2 w & = 10 \\3 x - 2 y + z - 4 w & = - 22 \\- 4 x + 3 y - 2 z + w & = - 2 \\- 2 x + y - 4 z + 3 w & = - 10\end{aligned} \right.$ ​

Select the order for the following matrix.​ $\left[ \begin{array} { r r r } - 9 & 8 & 7 \\0 & - 5 & 3\end{array} \right]$ ​

Determine whether the following matrix is in row-echelon form.If it is,determine if it is also in reduced row-echelon form. ​​ $\left[ \begin{array} { l l l l } 1 & 0 & 1 & 0 \\0 & 1 & 0 & 3 \\0 & 0 & 1 & 0\end{array} \right]$ ​

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { r r r r } 9 & - 6 & - 3 & - 33 \\8 & - 1 & 1 & - 47 \\- 7 & - 7 & 6 & - 8\end{array} \right]$

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}2 x + 9 y & = - 35 \\- 9 x + 2 y & = 30\end{aligned} \right.$

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.​ $\left\{ \begin{array} { c } 4 x + 9 y + z = - 17 \\6 x - 6 y - 8 z = 12 \\9 x + 8 y + z = - 43\end{array} \right.$

An augmented matrix that represents a system of linear equations (in variables x,y,z and w if applicable)has been reduced using Gauss-Jordan elimination.Find the solution represented by the augmented matrix.​ $\left[\begin{array}{lll}1 & 0 & \vdots 5\\0 & 1 & \vdots -6\\\end{array} \right]$ ​