REDUCTIONMETHOD052 - Chapter 6: MATRIX ALGEBRA Dr. Raja...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 6: MATRIX ALGEBRA Dr. Raja Mohammad Latif We will show how to reduce a matrix and to use matrix reduction to solve a linear system.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Math 131-052. Dr. Raja Latif 2 6.4-6.5: Method of Reduction Performing any one of the following Row Operations on the augmented matrix of a system of linear equations produces the augmented matrix of an equivalent system. 1. Interchange any two rows. 2. Multiply each element of a row by a nonzero constant. 3. Replace a row by the sum of itself and a constant multiple of another row of the matrix. DEPENDENT AND INCONSISTENT SYSTEMS. The possible number of solutions of a system with more than two variables or equations is the same as for smaller systems. Such a system has exactly one solution (independent Both the equation method and the matrix method always produce the unique solution of an independent system. REDUCED ROW ECHELON FORM: A system of m Department of Mathematics, KFUPM.
Background image of page 2
Math 131-052. Dr. Raja Latif 3 linear equations containing n variables will have either no can determine which of these possibilities occurs and, if the augmented matrix of the system until we arrive at the reduced row echelon form of the augmented matrix. Reduced Row Echelon Form of an Augmented Matrix: above it and below it. The leftmost 1 in any row is to the right of the leftmost 1 in the row above. Any rows that contain all 0s to the left of the vertical bar appear at the bottom. THE GAUSS-JORDAN METHOD FOR SOLVING A SYSTEM OF LINEAR EQUATIONS (Reduced Row Echelon Form of an Augmented Matrix) 1. Arrange the equations with the variable terms in the Department of Mathematics, KFUPM.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Math 131-052. Dr. Raja Latif 4 same order on the left of the equals sign and the constants on the right. 2. Write the augmented matrix that represents the system. 3. Perform row operations that place the entry 1 in row 1 , column 1 unchanged, while causing 0 s to appear below it in column 1 : i.e., Choose the leftmost nonzero column and use appropriate row operations to get a 1 at the top.) 4. Use multiples of the row containing the 1 from step 1 to get zeros in all remaining places in the column containing this 1 : 5. Perform row operations that place the entry 1 in row 2 ; column 2 ; but leave the entries in columns to the left unchanged. If it is impossible to place a 1 in row 2 ; column 2 ; then proceed to place a 1 in row 2 ; column 3 : Once a 1 is in place, perform row operations to place 0 s below it. Department of Mathematics, KFUPM.
Background image of page 4
Math 131-052. Dr. Raja Latif 5 (If any rows are obtained that contain only 0 s on the left side of the vertical bar, place such rows at the bottom of the matrix.) 6. Now repeat Step 4, placing a 1 in the next row, but one column to the right. Continue until the bottom row or the vertical bar is reached. 7. The matrix that results is the row echelon form of the
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

REDUCTIONMETHOD052 - Chapter 6: MATRIX ALGEBRA Dr. Raja...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online