Sec 3 2 Matrices and Gaussian Elemination Coefficient

Sec 3. 2 Matrices and Gaussian Elemination Coefficient Matrix 3 x 3 Augmented Coefficient Matrix 3 x 4

Sec 3. 2 Matrices and Gaussian Elemination Coefficient Matrix nxn Augmented Coefficient Matrix n x (n+1)

Elementary Row Operations 1 2 3 Multiply one equation by a nonzero constant Interchange two equations Add a constant multiple of one equation to another equation 1 2 3 Multiply one row by a nonzero constant Interchange two rows Add a constant multiple of one row to another row

1)Extra HW 2)Problem Session 3)Quiz 2 Stat 4)Chapter 1 Summ

How to solve any linear system Use sequence of elementary row operations Triangular system on i tut i t s ub e Us b ks c a

How to solve any linear system

Augmented Matrix A (-3) R 1 + R 2 (1/2) R 2 (-2) R 1 + R 3 (-3) R 2 + R 3 Definition: (Row-Equivalent Matrices) A and B are row equivalent if B can be obtained from A by a finite sequence of elementary row operations B Theorem 1: A and B are row equivalent A is the augmented matrix of sys(1) B is the augmented matrix of sys(2) & A and B are row equivalent sys(1) and sys(2) have same solution

Echelon Matrix zero row How many zero rows

Echelon Matrix non-zero row leading entry The first (from left) nonzero element in each nonzero row 1) How many non-zero rows 2) Find all leading entries

Echelon Matrix Def: A matrix A in row-echelon form if 1) All zero rows are at the bottom of the matrix 2) In consecutive nonzero rows the leading in the lower row appears to the right of the leading in the higher row

Echelon Matrix Transform each augmented matrix to echelon form. Then use back substitution to solve the system

Reduced Echelon Matrix Def: A matrix A in reduced-row-echelon form if 1) A is row-echelon form 2) All leading entries = 1 3) A column containing a leading entry 1 has 0’s everywhere else

Echelon Matrix Reduced Echelon Matrix 1) A row-echelon form 2) Make All leading entries = 1 (by division) 3) Use each leading 1 to clear out any nonzero elements in its column

Solving Linear System Gaussian Elimination Method: Solve: Gauss-Jordan Elimination Method: Row-echelon form Reduced Rowechelon form