1. 10.4 Matrix Algebra

2. 1. Matrix Notation A matrix is an array of numbers. Definition Definition: The Dimension of a matrix is m x n “m by n” where m = # rows, n = #columns A matrix is an array of numbers. Definition 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… To add/subtract… add corresponding elements. Evaluate: To add/subtract… 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 We can multiply matrix by a number (known as scalar). Example:Find

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

6. Evaluate Evaluate 4. Matrix Multiplication

7. more practice:

8. 4. Matrix Multiplication Dimensions: rows columns rows columns rows columns rows columns will have dimensions AB will have dimensionsDimensions: rows columns rows columns rows columns rows columns will have dimensions AB will have dimensions Important: For Matrix multiplication to work: The number of columns in first matrix must equal number of rows in second! 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 Real Numbers: 1 is the multiplicative identity. Example Matrices: 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. Real Numbers: 1 Real Numbers: 1 is the multiplicative identity. Example Matrices: 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 Real Numbers: Multiplicative Inverse of is (for any ) Matrices: Multiplicative Inverse Matrices: Multiplicative Inverse of a matrix is a matrix read as: “A-inverse” with the property: Real Numbers: Multiplicative Inverse Real Numbers: Multiplicative Inverse of is (for any ) Matrices: Multiplicative Inverse 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 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 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 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! 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: 2 nd MATRIX: Edit (Enter) Dimensions 3 x 3 To find Inverse: 2 nd MATRIX: NAMES 1:[A] Enter “^-1”. Example: Find the multiplicative inverse of Graphing calculator: To Enter Matrix data: 2 nd MATRIX: Edit (Enter) Dimensions 3 x 3 To find Inverse: 2 nd MATRIX: NAMES 1:[A] Enter “^-1”.

15. 7. Solve a system of linear equations Inverse Matrix method 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: 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: 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 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: Your turn: Solve the system: