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# 3.6 systems and matrices[1]

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### 3.6 systems and matrices[1]

1. 1. 3.6 SOLVING SYSTEMSUSING MATRICES
2. 2. MATRICES A matrix is a rectangular array of numbers, displayed within brackets. 2 4 1  A=  6 5 3   The dimensions of a matrix are the numbers of rows by the numbers of columns in the array.
3. 3. MATRICES Each number in a matrix is a matrix element and can be identified by its row and column number  Example:  a11 a12 a13  A=  a21 a22 a23  
4. 4. EXAMPLE: IDENTIFYING AMATRIX ELEMENTWhat is element a23 in matrix A?  4 −9 17 1 A= 0 5 8 6     −3 −2 10  0  
5. 5. SYSTEMS OF EQUATIONS ANDMATRICES We can represent systems of equations as matrices  Each row represents an equation  Each column represents the coefficients of a variable Example:
6. 6. REPRESENTING SYSTEMSWITH MATRICES
7. 7. EXAMPLE: REPRESENT THESYSTEM WITH A MATRIXx − 3y + z = 6 x + 3 z = 12 y = −5 x + 1
8. 8. EXAMPLE: WRITE THE SYSTEMOF EQUATIONS REPRESENTEDBY THE MATRIX5 2 7  0 1 9 
9. 9. SOLVING A SYSTEM USING AMATRIX We can solve a system by using a matrix and performing row operations Row Operations are the “legal moves and manipulations” we can make in a matrix Solving a system using row operations is similar to elimination, because we use the same steps, but don’t have variables
10. 10. SOLVING A SYSTEM USINGMATRICES Row Operations:  Switch any two rows  Multiply a row by a constant  Add (subtract) one row to another row Make sure you write down what you are doing!
11. 11. SOLVING A SYSTEM USINGMATRICES Goal:To use row operations to get a matrix in the following forms: 1 0 0 a  1 0 a      or 0 1 0 b  0 1 b  0 0 1 c     Matricesthat represent the solution of a system are in reduced row echelon form.
12. 12. SOLVE THE SYSTEM OFEQUATIONS USING A MATRIX x + 4 y = −12 x + 5 y = 4
13. 13. SOLVE THE SYSTEM OFEQUATIONS USING A MATRIX9 x − 2 y = 53 x + 7 y = 17
14. 14. SOLVE THE SYSTEM OFEQUATIONS USING A MATRIX x + 2 y = 163 x + y = 8
15. 15. ASSIGNMENT Page 179 #8 – 11, 13 – 23 odd, 24, 27 – 29
16. 16. 3.6 SOLVING SYSTEMSUSING MATRICESPart 2 – Three- Variable Systems
17. 17. USING MATRICES FOR THREEVARIABLE SYSTEMS Same goal and row operations used to solve a system with two variables
18. 18. SOLVING A SYSTEM USINGMATRICES Row Operations:  Switch any two rows  Multiply a row by a constant  Add (subtract) one row to another row Make sure you write down what you are doing!
19. 19. SOLVING A SYSTEM USINGMATRICES Goal:To use row operations to get a matrix in the following forms: 1 0 0 a  1 0 a      or 0 1 0 b  0 1 b  0 0 1 c     Matricesthat represent the solution of a system are in reduced row echelon form.
20. 20. SOLVE THE SYSTEM OFEQUATIONS USING A MATRIX
21. 21. SOLVE THE SYSTEM OFEQUATIONS USING A MATRIX
22. 22. SOLVE THE SYSTEM OFEQUATIONS USING A MATRIX
23. 23. ASSIGNMENT 3.6 Worksheet