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Matrices and Linear Systems

Matrices and Linear Systems. Key Vocabulary. Matrices-a rectangular arrangement of numbers that is used to organize information and solve problems. 6.1 Matrix Representations. Dimensions -the numbers of the rows and columns Entry or Element - each number in the matrix

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Matrices and Linear Systems

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  1. Matrices and Linear Systems

  2. Key Vocabulary • Matrices-a rectangular arrangement of numbers that is used to organize information and solve problems

  3. 6.1 Matrix Representations • Dimensions-the numbers of the rows and columns • Entry or Element- each number in the matrix • Identified as “a(subscript)ij” where iand j are the row number and column number column row

  4. 6.2 Matrix Operations Matrix Addition Add the corresponding entries *you can only add matrices that have the same dimension Scalar Multiplication Multiply the scalar by each value in the matrix Matrix Multiplication Multiply each entry in a row of matrix [A] by the corresponding entries in a column of matrix [B] *the number of entries in a row of matrix [A] must equal the number of entries in a column of matrix [B]

  5. Examples of Matrix Operations Matrix Addition Matrix Multiplication Scalar Multiplication

  6. 6.3 Solving Systems With Inverse Matrices • Identity Matrix is the square matrix that does not alter the entries of a square matrix under multiplication. [A] [I] = [A] • Inverse Matrix is the matrix that will produce an identity matrix when multiplied by [A]. [A] [A]^-1 = [1]

  7. Examples of Identity Matrix [ ] [ ] ] [ = 2 b 2 1 a 1 4 3 4 c 3 d ] 0 1 0 1 MULTIPLY EACH SIDE AND THEN USE SYSTEM OF EQUATIONS TO FIND EACH VARIABLE

  8. Example of Solving a System Using the Inverse Matrix • ] • ] • [ • [ • [ • ] • [ • = A X B -1 -1 • = ] [ I • -1 • = • -1 • =

  9. 6.4 Row Reduction Method 6.4 Row Reduction Method • Augmented Matrix is a single matrix that contains columns for the coefficients of each variable and a final column for the constant terms.

  10. 6.4 Row Reduction Method (cont.) • Similar to Elimination ] [ ORIGINAL EQUATION: 2x+y=5 5x+3y=13 5 2 1 0 .5 13 TO…. SOLUTION: (2,1)

  11. 6.5 Systems of Inequalities • Used to model real-world situations. • Systems of inequalities are similar to equations, however, when inequalities are multiplied or divided by a negative number the sign flips. • Inequalities are used to show restraints or limits for values. • To graph an equality, you graph the line and shade above or below the line depending on whether the value is greater than or less.

  12. 6.6 Linear Programming • Linear programming is finding a feasible region and finding the points that either give the maximum or minimum value to a specific expression. • This is used in business to find out ways of making maximum profit.

  13. Examples of Linear Programming • A pottery shop makes two kinds of pots, glazed and unglazed. An unglazed pot takes .5 hours to make on the pottery wheel, and 3 hours in the kiln. A glazed pot takes 1 hour on the wheel, and 18 in the kiln. The wheel is available for 10 hours a day. The 3 kilns in total can be used for 60 hours a day. The workshop must make at least 4 unglazed pots a day. The profit on an unglazed pot is 15$ and a glazed pot makes 50$. How many of each pot should the shop make to maximize profit?

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