1. CHAPTER 2 MATRICES 2.1 Operations with Matrices Matrix
(i, j)-th entry:
row: m
column: n
size: m×n

2. i-th row vector
row matrix
j-th column vector
column matrix
Square matrix: m = n

7. Matrix form of a system of linear equations:= A x b

8. Partitioned matrices:
submatrix

9. linear combination of column vectors of A
a linear combination of the column vectors of matrix A: = = =
linear combination of column vectors of A

10. 2.2 Properties of Matrix Operations
Three basic matrix operators:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication
Zero matrix:
Identity matrix of order n:

14. (Commutative law for multiplication)
Real number:
ab = ba
(Commutative law for multiplication)
Matrix:
Three situations:
(Sizes are not the same)
(Sizes are the same, but matrices are not equal)

15. Real number:
(Cancellation law)
Matrix:
(1) If C is invertible, then A = B
(Cancellation is not valid)

17. Transpose of a matrix:

20. A square matrix A is symmetric if A = AT
Symmetric matrix:
A square matrix A is symmetric if A = AT
Skew-symmetric matrix:
A square matrix A is skew-symmetric if AT = –A
Note:
is symmetric
Proof:

21. 2.3 The Inverse of a Matrix
Notes:

22. If A can’t be row reduced to I, then A is singular.

23. Power of a square matrix:

25. Note:

26. Note:
If C is not invertible, then cancellation is not valid.

28. 2.4 Elementary Matrices Note:
Only do a single elementary row operation.

32. Note:
If A is invertible

35. Note:
If a square matrix A can be row reduced to an upper triangular
matrix U using only the row operation of adding a multiple of
one row to another, then it is easy to find an LU-factorization of A.

36. Solving Ax=b with an LU-factorization of A
Two steps:
(1) Write y = Ux and solve Ly = b for y
(2) Solve Ux = y for x