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8.1 Matrices and Systems of Equations

8.1 Matrices and Systems of Equations. Let’s do another one:. we’ll keep this one. Now we’ll use the 2 equations we have with y and z to eliminate the y ’s. multiply the first equation by 8 and add the equations to eliminate y .

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8.1 Matrices and Systems of Equations

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  1. 8.1 Matrices and Systems of Equations

  2. Let’s do another one: we’ll keep this one Now we’ll use the 2 equations we have with y and z to eliminate the y’s.

  3. multiply the first equation by 8 and add the equations to eliminate y.

  4. Matrix: rectangular organization of real numbers in rows and columns Column 1 C 2 C 3 C n Row 1 Row 2 Row 3 Row m • Entry aijis a real number in a certain “i” row and “j” column

  5. Dimension, or ORDER, of a matrix • Dimension or order is given by m x n m = number of rows n = number of columns

  6. Square Matrix • In a square matrix: m = n; rows = columns • Note: To solve a system of equations, you will need square coefficient matrices

  7. Coefficient Matrix • A matrix whose real entries are the coefficients from a system of equations

  8. Solution Matrix • A column matrix whose entries are the solutions of the system of equations

  9. Augmented matrix- A matrix set up where the coefficient matrix and solution matrix of a system of equations is combinedNotation: Matrix square brackets with a bar separating the coefficients

  10. Augmented Matrix cont.’d

  11. Elementary Row Operations- Manipulating to Solve Matrices/ Systems 1.) Interchange two rows 2.) Multiply a row by a NONZERO multiple 3.) Add a multiplied row to another row

  12. Row- Echelon Form (ref) Page 548: 1.) If there is a row of all zeros, it is the last row 2.) The first nonzero entry of a nonzero row is 1 3.) The 1’s occur at a diagonal Reduced Row-echelon Form (rref) If every column that has a leading 1 has zeros above and below the 1.

  13. Page 548

  14. Gaussian Elimination 1.) Write the augmented matrix of a system of equations 2.) Use elementary row operations to rewrite the augmented matrix in ref form 3.) Rewrite the matrix as a system of equations and back-substitute in to solve for each variable

  15. Example 1: Solve Row 2: Added R1 + R2 -4 3 5 -1 3 -1 -3 0 -1 2 2 Row 3: Added -2(R1) + R3 -2 8 -6 -10 2 0 -4 6 0 8 -10 -4

  16. Row 3: Added 8(R2) + R3 0 -8 16 16 08 -10 -4 0 0 6 12 Row 3: 1/6(R2)

  17. No put back into a system of equations and back substitute

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