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10.4 Matrix Algebra

10.4 Matrix Algebra. 1. Matrix Notation. A matrix is an array of numbers. Definition : The Dimension of a matrix is m x n “m by n” where m = # rows, n = #columns. 2. Sum and Difference of 2 matrices. To add/subtract… add corresponding elements. Evaluate:.

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10.4 Matrix Algebra

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  1. 10.4 Matrix Algebra

  2. 1. Matrix Notation A matrix is an array of numbers. Definition: The Dimension of a matrix is m x n “m by n” where m = # rows, n = #columns

  3. 2. Sum and Difference of 2 matrices To add/subtract… add corresponding elements. Evaluate: Note: The matrices must be same dimensions!

  4. 3. Scalar Multiplication We can multiply matrix by a number (known as scalar). Example: Find

  5. 4. Matrix Multiplication Multiplication : Row-by-Column multiplication Determine

  6. 4. Matrix Multiplication Evaluate

  7. 4. Matrix Multiplication more practice:

  8. 4. Matrix Multiplication Dimensions: rows columns rows columns ABwill have dimensions Important: For Matrix multiplication to work: The number of columns in first matrix must equal number of rows in second! Why is the product BA not possible?

  9. 4. Matrix Multiplication Evaluate the following:

  10. 5. Identity Matrix Real Numbers: 1 is the multiplicative identity. Example Matrices: is the Multiplicative identity of a matrix , a square matrix with 1’s on diagonal, 0’s elsewhere. is used to represent the order n (dimension) Example: Order 2 Order 3 A matrix times its identity returns the original matrix.

  11. 6. Inverse of a Matrix Real Numbers: Multiplicative Inverse of is (for any ) Matrices: Multiplicative Inverse of a matrix is a matrix read as: “A-inverse” with the property: Definition: If a matrix does not have an inverse, it is called singular

  12. 6. Inverse of a Matrix Example: Given and its inverse show and

  13. 6. b) Finding the Inverse of a Matrix To find the inverse: 1) Form augmented matrix 2) Transform to reduced row echelon form (Gauss-Jordan). 3) The identity matrix will magically appear on the right hand side of the bar! This is Example: Find the multiplicative inverse of Verify it when finished!

  14. 6. b) Finding the Inverse of a Matrix • Example: • Find the multiplicative inverse of • Graphing calculator: • To Enter Matrix data: • 2nd MATRIX: Edit (Enter) • Dimensions 3 x 3 • To find Inverse: • 2nd MATRIX: NAMES 1:[A] Enter “^-1”.

  15. 7. Solve a system of linear equations Inverse Matrix method • A system can be written using matrix notation: • A is the coefficient matrix • B is the constant matrix • X represents the unknowns. • Example: Write this system using matrix notation:

  16. 7. Inverse matrix method If has a unique solution then is the solution. Solve:

  17. 7. Solve a linear system using inverse Matrix Example: Solve the system: Note: We found in an earlier example

  18. 7. Solve a linear system using inverse Matrix Your turn: Solve the system:

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