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Systems of Linear Equation and Matrices

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Systems of Linear Equation and Matrices

CHAPTER 1

FASILKOM UI 05

YR

- Information in science and mathematics is often organized into rows and columns to form rectangular arrays.
- Tables of numerical data that arise from physical observations
- Example: (to solve linear equations)
- Solution is obtained by performing appropriate operations on this matrix

Introduction to Systems

of Linear Equations

- In x y variables (straight line in the xy-plane)
where a1, a2, & b are real constants,

- In n variables
where a1, …, an & b are real constants

x1, …, xn = unknowns.

- The equations are linear (does not involve any products or roots of variables).

- The equations are not linear.
- A solution of is a sequence of n numbers s1, s2, ..., snЭ they satisfy the equation when x1=s1, x2=s2, ..., xn=sn (solution set).

- 1 equation and 2 unknown, set one var as the parameter (assign any value)
- or
- 1 equation and 3 unknown, set 2 vars as parameter

- Is A finite set of linear equations in the vars x1, ..., xn
- s1, ..., sn is called a solution if x1=s1, ..., xn=sn is a solution of every equation in the system.
- Ex.
- x1=1, x2=2, x3=-1the solution
- x1=1, x2=8, x3=1is not, satisfy only the first eq.
- System that has no solution : inconsistent
- System that has at least one solution: consistent

- Consider:

- (x,y) lies on a line if and only if the numbers x and y satisfy the equation of the line. Solution: points of intersection l1 & l2
- l1 and l2 may be parallel:
no intersection, no solution

- l1 and l2 may intersect
at only one point: one solution

- l1 and l2 may coincide:
infinite many points of intersection,

infinitely many solutions

- In general: Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions.
- An arbitrary system of m linear equations in n unknowns:
a11x1 + a12x2 + ... + a1nxn = b1

a21x1 + a22x2 + ... + a2nxn = b2

am1x1 + am2x2 + ... + amnxn = bm

- x1, ..., xn = unknowns, a’s and b’s are constants
- aij, i indicates the equation in which the coefficient occurs and j indicates which unknown it multiplies

- Example:
- Remark: when constructing, the unknowns must be written in the same order in each equation and the constants must be on right.

- Basic method of solving system linear equations
- Step 1: multiply an equation through by a nonzero constant.
- Step 2: interchange two equations.
- Step 3: add a multiple of one equation to another.

- On the augmented matrix (elementary row operations):
- Step 1: multiply a row through by a nonzero constant.
- Step 2: interchange two rows.
- Step 3: add a multiple of one equation to another.

- r2= -2r1 + r2
- r3 = -3r1 + r3

- r2 = ½ r2
- r3 = -3r2 + r3
- r3 = -2r3

- r1 = r1 – r2
- r1 = -11/2 r3 + r1
- r2 = 7/2 r3 + r2
- Solution:

Gaussian Elimination

- Reduced row-echelon form, a matrix must have the following properties:
- If a row does not consist entirely of zeros the the first nonzero number in the row is a 1 = leading 1
- If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.
- In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.
- Each column that contains a leading 1 has zeros everywhere else.

- A matrix that has the first three properties is said to be in row-echelon form.
- Example:
- Reduced row-echelon form:
- Row-echelon form:

- Step 1: Locate the leftmost non zero column
- Step 2: Interchange
r2↔ r1.

- Step 3: r1 = ½ r1.
- Step 4: r3 = r3 – 2r1.

- Step 5 : continue do all steps above until the entire matrix is in row-echelon form.
- r2 = -½ r2
- r3 = r3 – 5r2
- r3 = 2r3

- Step 6 : add suitable multiplies of each row to the rows above to introduce zeros above the leading 1’s.
- r2 = 7/2 r3 + r2
- r1 = -6r3 + r1
- r1 = 5r2 + r1

- 1-5 steps produce a row-echelon form (Gaussian Elimination). Step 6 is producing a reduced row-echelon (Gauss-Jordan Elimination).
- Remark: Every matrix has a unique reduced row-echelon form, no matter how the row operations are varied. Row-echelon form of matrix is not unique: different sequences of row operations can produce different row- echelon forms.

- Bring the augmented matrix into row-echelon form only and then solve the corresponding system of equations by back-substitution.
- Example:

Matrices and Matrix Operations

Inverses; Rules of Matrix Arithmetic

- ab = ba for real numbers a & b, but AB ≠ BA even if both AB & BA are defined and have the same size.
- Example:

Theorem: Properties of

A+B = B+A

A+(B+C) = (A+B)+C

A(BC) = (AB)C

A(B+C) = AB+AC

(B+C)A = BA+CA

A(B-C) = AB-AC

(B-C)A = BA-CA

a(B+C) = aB+aC

a(B-C) = aB-aC

Math Arithmetic

(Commutative law for addition)

(Associative law for addition)

(Associative for multiplication)

(Left distributive law)

(Right distributive law)

(a+b)C = aC+bC

(a-b)C = aC-bC

a(bC) = (ab)C

a(BC) = (aB)C

- Proof (d):
- Proof for both have the same size:
- Let size A be r x m matrix, B & C be m x n (same size).
- This makes A(B+C) an r x n matrix, follows that AB+AC is also an r x n matrix.

- Proof that corresponding entries are equal:
- Let A=[aij], B=[bij], C=[cij]
- Need to show that [A(B+C)]ij = [AB+AC]ij for all values of i and j.
- Use the definitions of matrix addition and matrix multiplication.

- Proof for both have the same size:

- Remark: In general, given any sum or any product of matrices, pairs of parentheses can be inserted or deleted anywhere within the expression without affecting the end result.

- A matrix, all of whose entries are zero, such as
- A zero matrix will be denoted by 0 or 0mxn for the mxn zero matrix. 0for zero matrix with one column.
- Properties of zero matrices:
- A + 0 = 0 + A = A
- A – A = 0
- 0 – A = -A
- A0 = 0; 0A = 0

- Square matrices with 1’s on the main diagonal and 0’s off the main diagonal, such as
- Notation: In = n x n identity matrix.
- If A = m x n matrix, then:
- AIn = A and InA = A

- Example:
- Theorem: If R is the reduced row-echelon form of an n x n matrix A, then either R has a row of zeros or R is the identity matrix In.

- Definition: If A & B is a square matrix and same size Э AB = BA = I, then A is said to be invertible and B is called an inverse of A. If no such matrix B can be found, then A is said to be singular.
- Example:

- Theorem:
- If B and C are both inverses of the matrix A, then B = C.
- If A is invertible, then its inverse will be denoted by the symbol A-1.

- The matrix
is invertible if ad-bc ≠ 0, in which case the inverse is given by the formula

- If B and C are both inverses of the matrix A, then B = C.

- Theorem: If A and B are invertible matrices of the same size, then AB is invertible and (AB)-1 = B-1A-1.
- 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.
- Example:

- If A is a square matrix, then we define the nonnegative integer powers of A to be
A0=IAn = AA...A(n>0)

n factors

- Moreover, if A is invertible, then we define the negative integer prowers to be A-n = (A-1)n = A-1A-1...A-1
n factors

- Theorem: Laws of Exponents
- If A is a square matrix, and r and s are integers, then ArAs = Ar+s = Ars
- If A is an invertible matrix, then
- A-1 is invertible and (A-1)-1 = A
- An is invertible and (An)-1 = (A-1)n for n = 0, 1, 2, ...
- For any nonzero scalar k, the matrix kA is invertible and (kA)-1 = 1/k A-1.

- Example:

- If A is a square matrix, m x m, and if
is any polynomial, then we define

- Example:

- Theorem: If the sizes of the matrices are such that the stated operations can be performed, then
- ((A)T)T = A
- (A+B)T = AT + BT and (A-B)T = AT – BT
- (kA)T = kAT, where k is any scalar
- (AB)T = BTAT

- The transpose of a product of any number of matrices is equal to the product of their transpose in the reverse order.

- Theorem: If A is an invertible matrix, then AT is also invertible and (AT)-1 = (A-1)T
- Example:

Elementary Matrices and a Method for Finding A-1

- Definition:
- An n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix In by performing a single elementary row operation.

- Example:
- Multiply the second row of I2 by -3.
- Interchange the second and fourth rows of I4.
- Add 3 times the third row of I3 to the first row.

- Theorem: (Row Operations by Matrix Multiplication)
- If the elementary matrix E results from performing a certain row operation on Im and if A is an m x n matrix, then the product of EA is the matrix that results when this same row operation is performed on A.

- Example:
- EA is precisely the same matrix that results when we add 3 times the first row of A to the third row.

- If an elementary row operation is applied to an identity matrix I to produce an elementary matrix E, then there is a second row operation that, when applied to E, produces I back again.
- Inverse operation

- Theorem: Every elementary matrix is invertible, and the inverse is also an elementary matrix.
- Theorem: (Equivalent Statements)
- If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false.
- A is invertible
- Ax = 0 has only the trivial solution.
- The reduced row-echelon form of A is In.
- A is expressible as a product of elementary matrices.

- If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false.

- Proof:
Assume A is invertible and let x0 be any solution of Ax=0.

Let Ax=0 be the matrix form of the system

Assumed that the reduced row-echelon form of A is In by a finite sequence of elementary row operations, such that:

By theorem, E1,…,En are invertible. Multiplying both sides of equation on the left we obtain:

This equation expresses A as a product of elementary matrices.

If A is a product of elementary matrices, then the matrix A is a product of invertible matrices, and hence is invertible.

- Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent.
- An n x n matrix A is invertible if and only if it is row equivalent to the n x n identity matrix.

- To find the inverse of an invertible matrix, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtain A-1.
- Example:
- Adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A|I]
- Apply row operations to this matrix until the left side is reduced to I, so the final matrix will have the form [I|A-1].

Added –2 times the first row to the second and –1 times the first row to the third.

Added 2 times the second row to the third.

Multiplied the third row by –1.

Added 3 times the third row to the second and –3 times the third row to the first.

We added –2 times the second row to the first.

- Often it will not be known in advance whether a given matrix is invertible.
- If elementary row operations are attempted on a matrix that is not invertible, then at some point in the computations a row of zeros will occur on the left side.
- Example: