1. Unit 3: Matrices

2. Determine the dimensions of each matrix.

3. Example
Identify each element.
a23
a12
a31
a21

4. Matrix Multiplication
When multiplying matrices A and B, the number of COLUMNS in matrix A MUST be equal to the ROWS in matrix B.
The size of the product is:
# rows in A x # columns in B.

5. How to multiply matrices
Multiply the elements of each row in the first matrix by the elements in each column of the second matrix
Add the products to get the new element.

7. Multiplying Matrices
Can the following Matrices be multiplied? If so, what dimensions will the product be?? x x
1x3 and 3x3
1x3

8. Matrix Multiplication

9. DETERMINANT of Matrices

10. Determinant of a Matrix
A special number that can be calculated from the matrix.
It tells us things about the matrix that are useful in systems of linear equations, in calculus, and more
The symbol for determinant is two vertical lines either side

11. Determinant of a 2x2

12. Find the determinant of the following 2x2 matrices:

13. Determinant of a 3x3 Matrix

14. Find the determinant of the following.
-161 and 139

15. Matrix Equations

16. Matrix Equation Example

17. Solve each equation:

19. Inverse of Matrices

20. For matrices, there is no such thing as division
For matrices, there is no such thing as division. You can add, subtract, and multiple matrices, but you cannot divide them.
There is a related concept called inversion

21. Using Inverses to Solve
AX=C

22. Inverse Notation
REMEMBER we denote inverse with a -1 power So the inverse of matrix A is A-1

23. Requirement to have an Inverse
Matrix MUST be square, meaning it has the same number of rows and columns Matrix MUST NOT have a determinant of zero.

24. Inverse exist?!
Does the inverse exist?!?!

25. Multiplying Inverse
When you Multiply a matrix A times it’s inverse, the Product is the Identity Matrix. Identity Matrix is a square matrix where the top left to Bottom right diagonal are all ones, and everything else is a zero

26. Determine if the following matrices are inverses. 1. 2.

27. Finding the Inverse of a 2x2IF
THEN

28. Find the inverse of the following matrix.

29. Use your calculator! 2nd  Matrix  Edit Put in your matrix
2nd  Matrix  NAME
Recall your matrix
X-1

30. The inverse of a matrix can be used when solving matrix equations
The inverse of a matrix can be used when solving matrix equations. For Matrices A and B, we can find Matrix X: IF AX = B THEN X = A-1B

31. *Solve for X:
X = A-1B

32. You Try! Solve Each Matrix Equation:

33. Solve each matrix equation.
Solutions: