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3.6 systems and matrices

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3.6 systems and matrices

1. 1. 3.6 SOLVING SYSTEMS USING MATRICES
2. 2. MATRICES <ul><li>A matrix is a rectangular array of numbers, displayed within brackets. </li></ul><ul><ul><li>The dimensions of a matrix are the numbers of rows by the numbers of columns in the array. </li></ul></ul>
3. 3. MATRICES <ul><li>Each number in a matrix is a matrix element and can be identified by its row and column number </li></ul><ul><ul><li>Example: </li></ul></ul>
4. 4. EXAMPLE: IDENTIFYING A MATRIX ELEMENT <ul><li>What is element in matrix A ? </li></ul>
5. 5. SYSTEMS OF EQUATIONS AND MATRICES <ul><li>We can represent systems of equations as matrices </li></ul><ul><ul><li>Each row represents an equation </li></ul></ul><ul><ul><li>Each column represents the coefficients of a variable </li></ul></ul><ul><li>Example: </li></ul>
6. 6. REPRESENTING SYSTEMS WITH MATRICES <ul><li>Stack the variables/constants </li></ul><ul><li>Write the matrix using coefficients and constants. </li></ul><ul><ul><li>Use a 1 if there is no coeffiecient </li></ul></ul><ul><ul><li>Use a 0 if the variable is not included in the equation </li></ul></ul>
7. 7. EXAMPLE: REPRESENT THE SYSTEM WITH A MATRIX
8. 8. EXAMPLE: WRITE THE SYSTEM OF EQUATIONS REPRESENTED BY THE MATRIX
9. 9. SOLVING A SYSTEM USING A MATRIX <ul><li>We can solve a system by using a matrix and performing row operations </li></ul><ul><li>Row Operations are the “legal moves and manipulations” we can make in a matrix </li></ul><ul><li>Solving a system using row operations is similar to elimination, because we use the same steps, but don’t have variables </li></ul>
10. 10. SOLVING A SYSTEM USING MATRICES <ul><li>Row Operations: </li></ul><ul><ul><li>Switch any two rows </li></ul></ul><ul><ul><li>Multiply a row by a constant </li></ul></ul><ul><ul><li>Add one row to another row </li></ul></ul>
11. 11. SOLVING A SYSTEM USING MATRICES <ul><li>Goal: To use row operations to get a matrix in the following forms: </li></ul><ul><ul><li>Matrices that represent the solution of a system are in reduced row echelon form . </li></ul></ul>
12. 12. SOLVE THE SYSTEM OF EQUATIONS USING A MATRIX
13. 13. SOLVE THE SYSTEM OF EQUATIONS USING A MATRIX
14. 14. SOLVE THE SYSTEM OF EQUATIONS USING A MATRIX