2. RECORD

4. COLLABORATE:
Discuss: Is the statement below correct? Try a 2x2 example.

5. Matrix operations : Transposition
properties of matrix
transposition

6. Example: Matrix equations

7. Powers of Matrices:
Must A be a square matrix?
k times

9. Partitioned Matrices:
partition lines (horizontal, vertical)
blocks of a partitioned matrix

10. Multiplication of Partitioned Matrices: block-wise multiplication
The column partition of A needs “to match” the row partition of B:
# of horizontal blocks in A = # of vertical blocks in B
# of columns in each horizontal block in A = # of rows in the corresponding vertical block of B 1

11. Multiplication of Partitioned Matrices:

12. Block-Diagonal Matrices:
a block-diagonal matrix:
square blocks are on the diagonal
zero blocks everywhere else
all square matrices

13. Multiplication of Block-Diagonal Matrices:

14. Special Forms of Matrices:
diagonal matrix
symmetric matrix
skew-symmetric matrix
the identity matrix

15. Special Forms of Matrices:
upper triangular
lower triangular

16. Special Forms of Matrices: Reduced Row Echelon Form (rref) or “row reduced form”
Row Echelon Form (ref):
Leading 1’s.
Leading 1’s are shifted to the right.
All zero rows are on the bottom.
Elements below leading 1’s are zero.

18. Systems of Linear Equations:
Examples:

19. Systems of Linear Equations: the matrix notation
Examples:
With the augmented matrix [A|b]

20. Systems of Linear Equations: number of solutions
2-space:
3-space:

21. Equivalent Elementary Operations: Systems and Matrices

22. Gaussian Elimination:
derived system
back-substitution

23. Exercise: Solve the system by Gaussian elimination:
derived system
back-substitution

24. Exercise: Solve the system by Gaussian elimination: