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. A matrix with m rows and n columns is called an m by n matrix. Notation Notation: The order (dimensions) may also be written m x n. Example: Given the Matrix 1.Identify 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 Notation: The order (dimensions) may also be written m x n. Example: Given the Matrix 1.Identify

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

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

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) 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 On you own… Review Section 10.4 p. 773 for a summary of the scalar multiplication properties

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

Evaluate Evaluate 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: On your own… Review Section 10.4 p. 777 for a summary of the multiplication properties On your own… Review Section 10.4 p. 777 for a summary of the multiplication properties 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!

4. Matrix Multiplication Your turn to practice:

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: Identity Property Example:

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 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 Definition: If a matrix does not have an inverse, it is called singular

6. a) Verifying the Inverse of a Matrix The Multiplicative Inverse of the matrix is, The Multiplicative Inverse of the matrix A is, “A inverse” and satisfies “A inverse” and satisfies Example: We can show the inverse of is We must show and The Multiplicative Inverse of the matrix is, The Multiplicative Inverse of the matrix A is, “A inverse” and satisfies “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 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 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 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! 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: