1. MATRICES MATRIX Multiplication

2. Warm-up Subtract (don’t forget to KCC):

3. Matrix Multiplication  Matrix Multiplication is NOT Commutative! Order matters!  You can multiply matrices only if the number of columns in the first matrix equals the number of rows in the second matrix. 2 columns 2 rows

4. Matrix Multiplication  Take the numbers in the first row of matrix #1. Multiply each number by its corresponding number in the first column of matrix #2. Total these products. The result, 11, goes in row 1, column 1 of the answer. Repeat with row 1, column 2; row 1 column 3; row 2, column 1;...

5. Matrix Multiplication  Notice the dimensions of the matrices and their product. 3 x 22 x 33 x 3 __

6. Matrix Multiplication  Another example: 3 x 22 x 13 x 1

7. Matrix Determinants  A Determinant is a real number associated with a matrix. Only SQUARE matrices have a determinant.  The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A| or.

8. Matrix Determinants To find the determinant of a 2 x 2 matrix, multiply diagonal #1 and subtract the product of diagonal #2. Diagonal 1 = 12 Diagonal 2 = -2

9. Matrix Determinants To find the determinant of a 3 x 3 matrix, first recopy the first two columns. Then do 6 diagonal products. -20 -24 36 18 6016

10. Matrix Determinants The determinant of the matrix is the sum of the downwards products minus the sum of the upwards products. -20 -24 36 18 6016 = (-8) - (94) = -102

11. Identity Matrices  An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else.  When you multiply a matrix by the identity matrix, you get the original matrix.

12. Inverse Matrices  When you multiply a matrix and its inverse, you get the identity matrix.

13. Inverse Matrices  Not all matrices have an inverse!  To find the inverse of a 2 x 2 matrix, first find the determinant. a)If the determinant = 0, the inverse does not exist!  The inverse of a 2 x 2 matrix is the reciprocal of the determinant times the matrix with the main diagonal swapped and the other terms multiplied by -1.

14. Inverse Matrices Example 1:

15. Inverse Matrices Example 2: