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8.1 Matrix Solutions to Linear Systems Veronica Fangzhu Xing 3 rd period

8.1 Matrix Solutions to Linear Systems Veronica Fangzhu Xing 3 rd period. Solving Linear System Using Matrices An augmented matrix has a vertical bar separating the columns of the matrix into two groups The coefficients of each variable -------- the left of the vertical line

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8.1 Matrix Solutions to Linear Systems Veronica Fangzhu Xing 3 rd period

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  1. 8.1 Matrix Solutions to Linear SystemsVeronicaFangzhuXing3rd period

  2. Solving Linear System Using Matrices • An augmented matrix has a vertical bar separating the columns of the matrix into two groups • The coefficients of each variable -------- the left of the vertical line The constants---------right ( if any variable is missing, its coefficient is 0) x +2y -5z =-19 y +3z =9 z =4

  3. Matrix Row Operations 

  4. Solving linear System Using Gaussian Elimination • Write the augmented matrix for the system. • Write the system of linear equations corresponding to the matrix in step 2 and use back-substitution to find the system’s solution.

  5. Example 3 : Use matrices to solve the system: 3x+y+2z=31 x+y+2z=19 x+3y+2z=25 • Step 1 : Write the augmented matrix for the system.

  6. Step 2 : Use matrix row operations to simplify the matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s.

  7. Step 3 : Write the system of linear equation corresponding to the matrix in step 2 and use back-substitution to find the system’s solution.

  8.  Solving linear system Using Gauss Jordan Elimination • 1. Write the augmented matrix for the system. • 2. Use matrix row operations to simplify the matrix to a row-equivalent matrix in reduced row-echelon form, with1sdown the main diagonal from upper left to lower right, and 0s above and below the 1s • Get 1 in the upper left-hand corner • Use the 1 in the first column to get 0s below it • Get 1 in the second row, second column. • Use the 1 in the second column to make the remaining entries in the second column 0 • Get 1 in the third row, third column. • Use the 1 in the third column to make the remaining entries in the third column 0. • Continue this procedure as far as possible. • 3. Use the reduced row-echelon form of the matrix step 2 to write the system’s solution set.( back-substitution is not necessary)

  9. Example 4 : Use Gauss-Jordan elimination to solve the system 3x+y+2z=31 x+y+2z=19 x+3y+2z=25

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