1. Reduced echelon form Matrix equations Null space Range Determinant Invertibility Similar matrices Eigenvalues Eigenvectors Diagonabilty Power

2. Reduced echelon form Because the reduced echelon form of A is the identity matrix, we know that the columns of A are a basis for R 2 Return to outline

3. Matrix equations Because the reduced echelon form of A is the identity matrix: Return to outline

5. Every vector in the range of A is of the form: Is a linear combination of the columns of A. The columns of A span R 2 = the range of A Return to outline

6. The determinant of A = (1)(7) – (4)(-2) = 15 Return to outline

7. Because the determinant of A is NOT ZERO, A is invertible (nonsingular) Return to outline

8. If A is the matrix for T relative to the standard basis, what is the matrix for T relative to the basis: Q is similar to A. Q is the matrix for T relative to the  basis, (columns of P) Return to outline

9. The eigenvalues for A are 3 and 5 Return to outline

12. A square root of A = A 10 = Return to outline

13. The reduced echelon form of B =

14. Return to outline

16. The range of B is spanned by its columns. Because its null space has dimension 2, we know that its range has dimension 2. (dim domain = dim range + dim null sp). Any two independent columns can serve as a basis for the range.

17. Return to outline Because the determinant is 0, B has no inverse. ie. B is singular

18. Return to outline If P is a 4x4 nonsingular matrix, then B is similar to any matrix of the form P -1 BP

19. Return to outline The eigenvalues are 0 and 2.

20. Return to outline The null space of (0I –B) = the null space of B. The eigenspace belonging to 0 = the null space of the matrix The null space of (2I –B)= The eigenspace belonging to 2

21. Return to outline There are not enough independent eigenvectors to make a basis for R 4. The characteristic polynomial root 0 is repeated three times, but the eigenspace belonging to 0 is two dimensional. B is NOT similar to a diagonal matrix.

22. Return to outline The reduced echelon form of C is

23. Return to outline

24. A basis for the null space is:

25. Return to outline The columns of the matrix span the range. The dimension of the null space is 1. Therefore the dimension of the range is 2. Choose 2 independent columns of C to form a basis for the range

26. Return to outline The determinant of C is 0. Therefore C has no inverse.

27. Return to outline For any nonsingular 3x3 matrix P, C is similar to P -1 CP

28. Return to outline The eigenvalues are: 1, -1, and 0

29. Return to outline Its null space =

30. Return to outline The columns of P are eigenvectors and the diagonal elements of D are eigenvalues.

31. Return to outline