1. Ch. 7 – Matrices and Systems of Equations 7.5 – Operations with Matrices

2. Matrices  What is the order (dimensions) of each matrix?  Augmented Matrix: a matrix derived from a system of equations 2 x 13 x 32 x 3 SystemAugmented Matrix (dotted line optional)

3. Evaluate: 1. ----> 2. ----> 3. ----> 4. ---->

4. Evaluate: 1. ----> 2. ----> 3. ----> 4. ---->

5. Multiplying Matrices  In order to multiply 2 matrices, you must do row-by-column multiplication to calculate entries for the solution matrix.  Ex: Find the product of the matrices at right.  Multiply the rows of the 1 st matrix by the column in the 2 nd matrix!  Since 22 is the product of row 1 of the 1 st matrix and column 1 of the 2 nd matrix, it will go in the 1 st row and 1 st column of the solution matrix  Since -14 is the product of row 2 of the 1 st matrix and column 1 of the 2 nd matrix, it will go in the 2 nd row and 1 st column of the solution matrix

6.  Ex: Find the product of the matrices at right. Do row-by-column multiplication! 1014 -11-16

7.  Ex: Find the product of the matrices at right. Do row-by-column multiplication!  Wait…we can’t do this because we can’t multiply a row of 3 entries by a column of 2 entries!  Matrix multiplication only works in the following situation:  For matrices A, B, and C, other properties of matrices include:  A(BC) = (AB)C  A(B + C) = AB + AC Equal Order of AB

8.  Ex: Find the product of the matrices at right. It’s a 3x1 times a 1x3…can you do it? 2 1230 0 0 8 -4 The answer will be a 3x3 matrix!

9. Evaluate: 1. ----> 2. ----> 3. ----> 4. Not possible