1. MATRIX ALGABRA
YULVI ZAIKA

2. Matrix Operations
Although we introduced matrices as a structure for convenient “bookkeeping” when solving systems of linear equations, they are interesting mathematically in their own right.
We can define the operations of addition, scalar multiplication, subtraction and multiplication on them.

3. Matrix Terminology
We say that two mxn matrices A and B are equal if they have the same size and ai,j=bi,j for 1≤ i ≤ m and 1 ≤ j ≤ n.
A matrix with only one row is called a row matrix or row vector.
A matrix with only one column is called a column matrix and column vector. (ai)

4. Matrix Addition
If A and B are two m x n matrices then A+B is the m x n matrix with entries
(a+b)i.j= ai,j+bi,j.
Matrik yang berbeda ukurannya tidak bisa dijumlahkan
Ini juga berlaku untuk pengurangan

5. Scalar Multiplication
If A is an m x n matrix and c is a scalar then cA is the m x n matrix with entries,
(ca)i,j = c*ai,j.
mengalikan semua komponen matrix dengan konstanta
With this definition we can define A-B to be
A+(-1)B.

6. Matrix Multiplication
If A is an mx p matrix and B is an p x n matrix then AB is the m x n matrix with entries,

7. Consider the system of equations,
If A is the coefficient matrix of this system x is the column matrix of variables and b is the column matrix of right hand side numbers then the system is expressed as
Ax = b.

8. Properties of Matrix Addition
A+B = B+A
A+(B+C)=(A+B)+C
(cd)A = c(dA)
1A=A
c(A+B) = cA+cB
(c+d)A = cA +dA
A+0mn=A
A+(-A) = 0mn
If cA=0mn then c=0 or A=0mn.
Commutative
Associative
Scalar Associative
Scalar identity
Scalar distributive 1
Scalar distributive 2
Additive identity
Additive Inverse
Scalar cancellation property

9. Properties of Matrix Multiplication
A(BC) = (AB)C
A(B+C) = AB +AC
(A+B)C = AC+BC
c(AB) = (cA)B=A(cB)
AIn = A
ImA = A
assuming A is m by n and all operations are defined.
Associative
Left distributive
Right Distributive
Scalar Associative
Multiplicative Identity

10. Problem

11. Matrices Product as Linear Combination
Sometimes it is also useful to think of a linear system in the following way:

12. Matrixs Form of a linear system
A : coefficients of matrix
The augmented matrix

13. TRANPOSE MATRIX

14. In the special case where A is a square matrix, the transpose of A can be obtained by interchanging entries that are symmetrically positioned about the main diagonal

15. Trace of a Matrix

16. PROBLEM
Indicate whether the statement is always true or sometimes false. Justify your answer with a logical argument or a counterexample.
(a) The expressions tr (AAT) and tr (AT A)are always defined, regardless of the size of A.
(b) tr (AAT) = tr (AT A)for every matrix A.
(c) If the first column of A has all zeros, then so does the first column of every product AB.
(d) If the first row of A has all zeros, then so does the first row of every product AB.

17. Suppose the array
represents the orders placed by three individuals at a fast-food restaurant. The first person orders 4 burgers, 3 sodas, and 3 fries; the second orders 2 burgers and 1 soda, and the third orders 4 burgers, 4 sodas, and 2 fries. Burgers cost $2 each, sodas $1 each, and fries $1.50 each.
(a) Argue that the amounts owed by these persons may be represented as a function y=f(x)
, where f(x) is equal to the array given above times a certain vector.
(b) Compute the amounts owed in this case by performing the appropriate multiplication.
(c) Change the matrix for the case in which the second person orders an additional soda
and 2 fries, and recompute the costs.

18. PROPERTIES MATRIX OPERATION
1. AB and BA Need Not Be Equal
Thus, AB  BA

19. 2. Associativity of Matrix Multiplication; (AB)C= A(BC)

20. 3. Zero matrix

21. 4. Identity Matrix

22. Inverse matrix
The matrix with no inverse
The third column
Of BA is

23. PROPERTIES OF INVERSE
If B and C are both inverses of the matrix A, then B=C
Proof : Since B is an inverse of A, we have BA=I. Multiplying both sides on the right by C gives (BA)C=IC=C. But (BA)C=B(AC)=BI=B, so B=C .
As a consequence of this important result, we can now speak of “the” inverse of an invertible matrix. If A is invertible, then its
inverse will be denoted by the symbol A-1. Thus

24. A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order.

26. POWER OF MATRIX
If A is a square matrix, then we define the nonnegative integer powers of A to be
Moreover, if A is invertible, then we define the negative integer powers to be

27. Laws of Exponents (a) A-1 is invertible and (A-1 )-1=A .
If A is a square matrix and r and s are integers, then
If A is an invertible matrix, then:
(a) A-1 is invertible and (A-1 )-1=A .
(b) An is invertible and (An )-1= (A-1)n for n=0.1.2,.. .
(c) For any nonzero scalar k, the matrix kA is invertible and (kA)-1= 1/k A-1

29. Polynomial Expressions Involving Matrices
If A is a square matrix, say m x m, and if
is any polynomial, then we define

30. Properties of the Transpose
If the sizes of the matrices are such that the stated operations can be performed, then