2.5 - Determinants & Multiplicative Inverses of Matrices

1. 2.5 - Determinants & Multiplicative Inverses of Matrices

2. DETERMINANT • a real number representation of a square matrix. • The determinant of is a number denoted as or det • a matrix with a nonzero determinant is called nonsingular

3. Second-Order Determinant • The value of det or • is ad - cb.

4. 0(-6) - 8(-2) = 16 8(6) - 7(4) = 20 Examples • Find the value of • Find the value of

5. page 98

6. Third-Order Determinant

7. Find the value of

8. Option 2 for finding 140 + 0 + 18- 0 - -120 - 126 = 152

9. The Identity Matrix • a square matrix whose elements in the main diagonal, from upper left to lower right, are 1s, while all other elements are 0s.

10. Inverse Matrix • the product of a matrix and it’s inverse produces the identity matrix • only for square matrices • The inverse of matrix A would be denoted as A-1

11. Inverse of a Second-Order Matrix • First, the matrix must be nonsingular! • Then, if the matrix is nonsingular, an inverse exists. • If the detA = 0, then it is singular and no inverse exists.

12. Inverse of a Second-Order Matrix • If A = and , • then A-1 =

13. 1st - find the det 8(-1) - 3(9) = -35 2nd - find the inverse or Find the inverse of

14. DAY 2

15. Let’s use some technology! • it is important that you know how to do all these operations by hand. • matrices bigger than a second order are time consuming and well as multiplying matrices. • your calculators do all of this, but remember you will have a non-calculator section of your test.

17. are solving systems and matrices in the same chapter? You can use inverse matrices to solve systems of linear equations!

18. If we rewrite the system • as a product of matrices: Now, if this were a simple linear equation, like 5x = 15, how would you “get rid of” the 5?

19. First, find the inverse of • Then, multiply both sides by the inverse. (5, 2)

20. Use inverse matrices to solve