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Matrices

Matrices. Matrix - a rectangular array of variables or constants in horizontal rows and vertical columns enclosed in brackets. Element - each value in a matrix; either a number or a constant. Dimension - number of rows by number of columns of a matrix.

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Matrices

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  1. Matrices

  2. Matrix - a rectangular array of variables or constants in horizontal rows and vertical columns enclosed in brackets. Element - each value in a matrix; either a number or a constant. Dimension - number of rows by number of columns of a matrix. **A matrix is named by its dimensions.

  3. Examples: Find the dimensions of each matrix. Dimensions: 3x2 Dimensions: 4x1 Dimensions: 2x4

  4. Different types of Matrices • Column Matrix - a matrix with only one column. • Row Matrix - a matrix with only one row. • Square Matrix - a matrix that has the same number of rows and columns.

  5. Equal Matrices - two matrices that have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix. *The definition of equal matrices can be used to find values when elements of the matrices are algebraic expressions.

  6. Examples: Find the values for x and y * Since the matrices are equal, the corresponding elements are equal! * Form two linear equations. * Solve the system using substitution.

  7. 2. Set each element equal and solve!

  8. Matrix Operations • Addition • Subtraction • Multiplication • Inverse

  9. Addition

  10. Addition

  11. Addition Conformability To add two matrices A and B: • # of rows in A = # of rows in B • # of columns in A = # of columns in B

  12. Subtraction

  13. Subtraction

  14. Subtraction Conformability • To subtract two matrices A and B: • # of rows in A = # of rows in B • # of columns in A = # of columns in B

  15. Multiplication Conformability • Regular Multiplication • To multiply two matrices A and B: • # of columns in A = # of rows in B • Multiply: A (m x n) by B (n by p)

  16. Multiplication General Formula

  17. Multiplication I

  18. Multiplication II

  19. Multiplication III

  20. Multiplication IV

  21. Multiplication V

  22. Multiplication VI

  23. Multiplication VII

  24. Inner Product of a Vector • (Column) Vector c (n x 1)

  25. Outer Product of a Vector • (Column) vector c (n x 1)

  26. Inverse of 2 x 2 matrix • Find the determinant = (a11 x a22) - (a21 x a12) For det(A) = (2x3) – (1x5) = 1

  27. Inverse of 2 x 2 matrix • Swap elements a11 and a22 Thus becomes

  28. Inverse of 2 x 2 matrix • Change sign of a12 and a21 Thus becomes

  29. Inverse of 2 x 2 matrix • Divide every element by the determinant Thus becomes (luckily the determinant was 1)

  30. Inverse of 2 x 2 matrix • Check results with A-1 A = I Thus equals

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