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10.4 Matrix Algebra

10.4 Matrix Algebra. Matrix Notation Sum/Difference of 2 matrices Scalar multiple P roduct of 2 matrices Identity Matrix Inverse of a matrix a) Verify the inverse of a matrix b)Finding the inverse 7. S olve a system using inverse matrices. 1. Matrix Notation. Notation :

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10.4 Matrix Algebra

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  1. 10.4 Matrix Algebra Matrix Notation Sum/Difference of 2 matrices Scalar multiple Product of 2 matrices Identity Matrix Inverse of a matrix a) Verify the inverse of a matrix b)Finding the inverse 7. Solve a system using inverse matrices

  2. 1. Matrix Notation Notation: refers to the element in row i, column j of a matrix A. A matrix with m rows and n columns is called an m by n matrix. Notation: The order (dimensions) may also be written m x n. Example: Given the Matrix Identify

  3. 2. Sum and Difference of 2 matrices To add/subtract… We add corresponding elements. Evaluate: Note: The matrices must be same dimensions! If what is A + C ?

  4. 2. Sum and Difference of 2 matrices Properties The properties of addition also hold for matrix addition. On you own… Review Section 10.4 p. 770-772 for a summary of the properties

  5. 3. Scalar Multiplication We can multiply matrix by a number (known as scalar). kA implies the number k is multiplied times every element in A: Example: Find 1) 2) 3) 4) On you own… Review Section 10.4 p. 773 for a summary of the scalar multiplication properties

  6. 4. Matrix Multiplication Multiplication is NOT like addition (where we added corresponding elements). You will NOTmultiply corresponding elements. Matrix multiplication is performed row-by-column :

  7. 4. Matrix Multiplication Evaluate

  8. 4. Matrix Multiplication On your own… Review Section 10.4 p. 777 for a summary of the multiplication properties rows columns rows columns Example: is not possible when columns in A does not equal rows in B: Important: Matrix multiplication can only be performed if The number of columns in first matrix is equal to number of rows in second!

  9. 4. Matrix Multiplication Your turn to practice:

  10. 5. Identity Matrix Definition: The identity Matrix is a square matrix that has 1’s on diagonal and 0’s elsewhere An identity matrix has the same properties as 1 in the real numbers.

  11. 5. Identity Matrix Identity Property Example:

  12. 6. Inverse of a Matrix The Inverse is the matrix A is , “A inverse” and satisfies Example: We can show the inverse of is We must show and Definition: If a matrix does not have an inverse, it is called singular

  13. 6. a) Verifying the Inverse of a Matrix The MultiplicativeInverse of the matrix A is , “A inverse” and satisfies Example: We can show the inverse of is We must show and Definition: If a matrix does not have an inverse, it is called singular

  14. 6. b) Finding the Inverse of a Matrix To find the inverse: 1) Form the 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!

  15. 6. b) Finding the Inverse of a Matrix Example: Find the multiplicative inverse of Verify when finished! Your turn… Find the inverse for

  16. 7. Solve a system of linear equations using the inverse matrix method If a system has a unique solution where A is the coefficient matrix, X and B are 1 column matrices. then is the solution. Find Multiply The result in 2) is the solution

  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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