10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix a) Verify the inverse of a matrix b)Finding the inverse 7. Solve a system using inverse matrices

1. Matrix Notation Notation Notation: refers to the element in row i, column j of a matrix A. Notation Notation: An “ m x n” matrix has m rows and n columns Example: Identify the element Notation Notation: refers to the element in row i, column j of a matrix A. Notation Notation: An “ m x n” matrix has m rows and n columns Example: Identify the element

2. Sum and Difference of 2 matrices To add/subtract… To add/subtract… add corresponding elements. 1) 2) To add/subtract… To add/subtract… add corresponding elements. 1) 2) Note: The matrices must be same dimensions!

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) 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)

4. Matrix Multiplication Multiplication is Multiplication is NOT like addition (where we added corresponding elements). You will NOT multiply corresponding elements. Given: Find the product: Multiplication is Multiplication is NOT like addition (where we added corresponding elements). You will NOT multiply corresponding elements. Given: Find the product:

Evaluate Evaluate 4. Matrix Multiplication

Your turn to practice:

4. Matrix Multiplication rows columns rows columns rows columns rows columns Example: is not possible when columns in A does not equal rows in B: rows columns rows columns 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! Important: Matrix multiplication can only be performed if The number of columns in first matrix is equal to number of rows in second!

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. 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.

5. Identity Matrix Identity Property Example: Given the matrix: Identity Property Example: Given the matrix:

6. Inverse of a Matrix The Inverse is the matrix A is and satisfies Example: Given and its inverse show and The Inverse is the matrix A is and satisfies Example: Given and its inverse show and 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

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!

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

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. 1)Find 2)Multiply 1)The result in 2) is the solution If a system has a unique solution where A is the coefficient matrix, X and B are 1 column matrices. then is the solution. 1)Find 2)Multiply 1)The result in 2) is the solution

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

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