1. Goal: Find sums, differences, products, and inverses of matrices.
7.2
Matrix Algebra

2. What you’ll learn about
Matrices
Matrix Addition and Subtraction
Matrix Multiplication
Inverse Matrices
Determinant of a Square Matrix
… and why
Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

3. a rectangular arrangement of objects
each object in a matrix
rows x columns

4. Use the matrix at the right to answer the following questions.
Give the order of the matrix.
Identify the element in row 1 column 2.
If aij represents the element in row i column j, what is a31?
∆ ! % 𝜋 ∞ 𝛽

5. In the first three rounds of the Mens’ NCAA basketball tournament, Dayton scored 60, 55, and 82 points per game Arrange this data into two possible matrices.

6. Example: Determining the Order of a Matrix
Determine the order of each of the matrices.

7. Example: Identifying Elements
Identify the elements 𝑎 12 and 𝑎 32 in each matrix.

8. Example: Matrix Addition

9. Example: Using Scalar Multiplication

10. Matrix Multiplication

11. Example: Matrix Multiplication

12. Multiply. (Example #1)

13. Multiply. (Example #2)

14. Multiply. (Example #3)

15. Multiply and check your result on the calculator.

16. identity matrix

18. Find MI and IM. What do you notice?

19. Example: Solve for missing elements.
Solve for a and b.

20. Inverse of a 2 × 2 Matrix
An n  n matrix A has an inverse if and only if
det A ≠ 0.

21. Example: Finding Inverse Matrices
Let 𝐴=
Calculate det A and 𝐴 −1 .

22. Example: Finding Inverse Matrices

23. Example: Properties of Matrices

24. Properties of Matrices
Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.
1. ___________________property
Addition: A + B = B + A
Multiplication: Does not hold in general
2. ___________________property
Addition: (A + B) + C = A + (B + C)
Multiplication: (AB)C = A(BC)
3. ___________________ property
Addition: A + 0 = A
Multiplication: A·In = In·A = A

25. Properties of Matrices
Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.
4. ___________________ property
Addition: A + (-A) = 0
Multiplication: AA-1 = A-1A = In |A|≠0
5. ___________________ property
Multiplication over addition:
A(B + C) = AB + AC (A + B)C = AC + BC
Multiplication over subtraction:
A(B – C) = AB – AC (A – B)C = AC – BC