Sec 3.2 Matrices and Gaussian Elemination Coefficient Matrix 3 x 3 Coefficient Matrix 3 x 3 Augmented Coefficient Matrix 3 x 4 Augmented Coefficient Matrix 3 x 4

Sec 3.2 Matrices and Gaussian Elemination Coefficient Matrix n x n Coefficient Matrix n x n Augmented Coefficient Matrix n x (n+1) Augmented Coefficient Matrix n x (n+1)

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

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 Use back substitution

How to solve any linear system

(-3) R1 + R2 (-2) R1 + R3 (-3) R2 + R3 Augmented Matrix (1/2) R2 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 A B A and B are row equivalent A is the augmented matrix of sys(1) B is the augmented matrix of sys(2) Theorem 1: 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 1)How many non-zero rows 2)Find all leading entries leading entry The first (from left) nonzero element in each nonzero row

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

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 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 Echelon Matrix  Reduced Echelon Matrix

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