1. Matrices and Systems of Equations
Chapter 1
Matrices and Systems of Equations

2. Systems of Linear Equations
Where the aij’s and bi’s are all real numbers, xi’s are variables . We will refer to systems of the form (1) as m×n linear systems.

3. Definition Example Inconsistent : A linear system has no solution.
Consistent : A linear system has at least one solution.
Example
(ⅰ) x1 + x2 = 2
x1 − x2 = 2
(ⅱ) x1 + x2 = 2
x1 + x2 =1
(ⅲ) x1 + x2 = 2
−x1 − x2 =-2

4. Three Operations that can be used on a system
Definition
Two systems of equations involving the same variables are said to be equivalent if they have the same solution set.
Three Operations that can be used on a system
to obtain an equivalent system:
Ⅰ. The order in which any two equations are written may
be interchanged.
Ⅱ. Both sides of an equation may be multiplied by the
same nonzero real number.
Ⅲ. A multiple of one equation may be added to (or subtracted from) another.

5. is in strict triangular form.
n×n Systems
Definition
A system is said to be in strict triangular form if in the kth
equation the coefficients of the first k-1 variables are all
zero and the coefficient of xk is nonzero (k=1, …,n).
Example The system
is in strict triangular form.

6. Example Solve the system

7. Elementary Row Operations:
Ⅰ. Interchange two rows.
Ⅱ. Multiply a row by a nonzero real number.
Ⅲ. Replace a row by its sum with a multiple of another row.
Example Solve the system

8. 2 Row Echelon Form
pivotal row
pivotal row

9. Definition A matrix is said to be in row echelon form
ⅰ. If the first nonzero entry in each nonzero row is 1.
ⅱ. If row k does not consist entirely of zeros, the number of leading zero entries in row k+1 is greater than the number of leading zero entries in row k.
ⅲ. If there are rows whose entries are all zero, they are below the rows having nonzero entries.

10. Example Determine whether the following matrices are
in row echelon form or not.

11. Definition Definition
The process of using operations Ⅰ, Ⅱ, Ⅲ to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination.
Definition
A linear system is said to be overdetermined if there are more equations than unknows.
A system of m linear equations in n unknows is said to be underdetermined if there are fewer equations than unknows (m<n).

12. Example

13. Definition A matrix is said to be in reduced row echelon form if:
ⅰ. The matrix is in row echelon form.
ⅱ. The first nonzero entry in each row is the only nonzero entry in its column.

14. Homogeneous Systems
A system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero.
Theorem An m×n homogeneous system of linear equations has a nontrivial solution if n>m.

15. 3 Matrix Algebra
Matrix Notation

16. Vectors
row vector
1×n matrix
n×1 matrix
column vector

17. Scalar Multiplication
Definition
Two m×n matrices A and B are said to be equal if aij=bij for each i and j.
Scalar Multiplication
If A is a matrix and k is a scalar, then kA is the matrix
formed by multiplying each of the entries of A by k.
Definition
If A is an m×n matrix and k is a scalar, then kA is the m×n matrix whose (i, j) entry is kaij.

18. Matrix Addition Definition
Two matrices with the same dimensions can be added
by adding their corresponding entries.
Definition
If A=(aij) and B=(bij) are both m×n matrices,then the sum A+B is the m×n matrix whose (i, j) entry is aij+bij for each ordered pair (i, j).

19. Example
Let
Then calculate 。

20. cij = ai1b1j + ai2b2j +…+ ainbnj =  aikbkj.
Matrix Multiplication
Definition
If A=(aij) is an m×n matrix and B=(bij) is an n×r matrix, then the product AB=C=(cij) is the m×r matrix whose entries are defined by
cij = ai1b1j + ai2b2j +…+ ainbnj =  aikbkj.
k=1 n

21. Example
then calculate AB.
1. If
then calculate AB and BA.
2. If

22. Matrix Multiplication and Linear Systems
Case 1 One equation in Several Unknows
If we let and
then we define the product AX by

23. Case 2 M equations in N Unknows
If we let and
then we define the product AX by

24. Definition
If a1, a2, … , an are vectors in Rm and c1, c2, … , cn are scalars, then a sum of the form
c1a1+c2a2+‥‥cnan
is said to be a linear combination of the vectors a1, a2, … , an .
Theorem (Consistency Theorem for Linear Systems)
A linear system AX=b is consistent if and only if b can be written as a linear combination of the column vectors of A.

25. Theorem Each of the following statements is valid for any scalars k and l and for any matrices A, B and C for which the indicated operations are defined.
A+B=B+A
(A+B)+C=A+(B+C)
(AB)C=A(BC)
A(B+C)=AB+AC
(A+B)+C=AC+BC
(kl)A=k(lA)
k(AB)=(kA)B=A(kB)
(k+l)A=kA+lA
k(A+B)=kA+kB

26. The Identity Matrix
Definition
The n×n identity is the matrix where

27. Matrix Inversion Definition Definition
An n×n matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I.
Then matrix B is said to be a multiplicative inverse of A.
Definition
An n×n matrix is said to be singular if it does not have a multiplicative inverse.

28. The Transpose of a Matrix
Theorem If A and B are nonsingular n×n matrices, then AB is also nonsingular and (AB)-1=B-1A-1
The Transpose of a Matrix
Definition
The transpose of an m×n matrix A is the n×m matrix B defined by
bji=aij
for j=1, …, n and i=1, …, m. The transpose of A is denoted by AT.

29. Algebra Rules for Transpose:
(AT)T=A
(kA)T=kAT
(A+B)T=AT+BT
(AB)T=BTAT
Definition
An n×n matrix A is said to be symmetric if AT=A.

30. 4. Elementary Matrices
If we start with the identity matrix I and then perform exactly one elementary row operation, the resulting matrix is called an elementary matrix.

31. Type I. An elementary matrix of type I is a matrix obtained by
interchanging two rows of I.
Example Let
and let A be a 3×3 matrix
then

32. Type II. An elementary matrix of type II is a matrix obtained by
multiplying a row of I by a nonzero constant.
Example Let
and let A be a 3×3 matrix
then

33. Type III. An elementary matrix of type III is a matrix obtained
from I by adding a multiple of one row to another row.
Example Let
and let A be a 3×3 matrix

34. In general, suppose that E is an n×n elementary matrix
In general, suppose that E is an n×n elementary matrix. E is obtained by either a row operation or a column operation.
If A is an n×r matrix, premultiplying A by E has the
effect of performing that same row operation on A. If B
is an m×n matrix, postmultiplying B by E is equivalent
to performing that same column operation on B.

35. Example
Let ,
Find the elementary matrices ， ，such that

36. Theorem If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type.
Definition
A matrix B is row equivalent to A if there exists a finite sequence E1, E2, … , Ek of elementary matrices such that
B=EkEk-1‥‥E1A

37. Theorem 1.4.2 (Equivalent Conditions for Nonsingularity)
Let A be an n×n matrix. The following are equivalent:
A is nonsingular.
Ax=0 has only the trivial solution 0.
A is row equivalent to I.
Theorem The system of n linear equations in n unknowns Ax=b has a unique solution if and only if A is nonsingular.

38. A method for finding the inverse of a matrix
If A is nonsingular, then A is row equivalent to I and
hence there exist elementary matrices E1, …, Ek such
that
EkEk-1‥‥E1A=I
multiplying both sides of this
equation on the right by A-1
EkEk-1‥‥E1I=A-1
row operations
Thus (A I)
(I A-1)

39. Example Compute A-1 if

40. Example Solve the system

41. Diagonal and Triangular Matrices
An n×n matrix A is said to be upper triangular if aij=0 for i>j
and lower triangular if aij=0 for i<j.
A is said to be triangular if it is either upper triangular or
lower triangular.
An n×n matrix A is said to be diagonal if aij=0 whenever i≠j .

42. 5. Partitioned Matrices -2 4 1 3 1 1 1 1 3 2 -1 2 = C= 4 6 2 2 4
C11 C12 =
C21 C22 C= B=
=(b1, b2, b3)
AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3)

43. In general, if A is an m×n matrix and B is an n×r that has
been partitioned into columns (b1, …, br), then the block
multiplication of A times B is given by
AB=(Ab1, Ab2, … , Abr)
If we partition A into rows, then
Then the product AB can be partitioned into rows as follows:

44. Case 1 B=(B1 B2), where B1 is an n×t matrix and B2
Block Multiplication
Let A be an m×n matrix and B an n×r matrix.
Case 1 B=(B1 B2), where B1 is an n×t matrix and B2
is an n×(r-t) matrix.
AB= A(b1, … , bt, bt+1, … , br)
= (Ab1, … , Abt, Abt+1, … , Abr)
= (A(b1, … , bt), A(bt+1, … , br))
= (AB1 AB2)

45. Case 2 A= ,where A1 is a k×n matrix and A2
is an (m-k)×n matrix.
Thus
Case 3 A=(A1 A2) and B= , where A1 is an m×s matrix
and A2 is an m×(n-s) matrix, B1 is an s×r matrix and B2 is an
(n-s)×r matrix.
Thus

46. Case 4 Let A and B both be partitioned as follows：
B11 B s B=
B21 B n-s
t r-t
A11 A k A=
A21 A m-k
s n-s
Then

47. In general, if the blocks have the proper dimensions, the block multiplication can be carried out in the same manner as ordinary matrix multiplication.

48. Example
Let
Then calculate AB.

49. Example
Let A be an n×n matrix of the form
where A11 is a k×k matrix (k<n). Show that
A is nonsingular if and only if A11 and A22
are nonsingular.