Systems of Linear Equation and Matrices

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Systems of Linear Equation and Matrices. CHAPTER 1 FASILKOM UI 05. YR. Introduction ~ Matrices. 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

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### Systems of Linear Equation and Matrices

CHAPTER 1

FASILKOM UI 05

YR

Introduction ~ Matrices
• 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

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.

• Example 1 Linear Equations
• The equations are linear (does not involve any products or roots of variables).
Linear Equations
• 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).
• Example 2 Finding a 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
Linear Systems / System of Linear Equations
• 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=-1 the solution
• x1=1, x2=8, x3=1 is not, satisfy only the first eq.
• System that has no solution : inconsistent
• System that has at least one solution: consistent
• Consider:
Linear Systems
• (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

Linear Systems
• 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
Augmented Matrices
• Example:
• Remark: when constructing, the unknowns must be written in the same order in each equation and the constants must be on right.
Augmented Matrices
• 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.
Elementary Row Operations (Example)
• r2= -2r1 + r2
• r3 = -3r1 + r3
Elementary Row Operations (Example)
• r2 = ½ r2
• r3 = -3r2 + r3
• r3 = -2r3
Elementary Row Operations (Example)
• r1 = r1 – r2
• r1 = -11/2 r3 + r1
• r2 = 7/2 r3 + r2
• Solution:
Echelon Forms
• 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.
Echelon Forms
• A matrix that has the first three properties is said to be in row-echelon form.
• Example:
• Reduced row-echelon form:
• Row-echelon form:
Elimination Methods
• Step 1: Locate the leftmost non zero column
• Step 2: Interchange

r2↔ r1.

• Step 3: r1 = ½ r1.
• Step 4: r3 = r3 – 2r1.
Elimination Methods
• Step 5 : continue do all steps above until the entire matrix is in row-echelon form.
• r2 = -½ r2
• r3 = r3 – 5r2
• r3 = 2r3
Elimination Methods
• 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
Elimination Methods
• 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.
Back-substitution
• Bring the augmented matrix into row-echelon form only and then solve the corresponding system of equations by back-substitution.
• Example:
Properties of Matrix Operations
• 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

(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

Properties of Matrix Operations
Properties of Matrix Operations
• 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.
Properties of Matrix Operations
• 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.
Zero Matrices
• 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
Identity Matrices
• 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
Identity Matrices
• 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.
Identity Matrices
• 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:
Properties of Inverses
• 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

Properties of Inverses
• 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:
Powers of a Matrix
• If A is a square matrix, then we define the nonnegative integer powers of A to be

A0=I An = 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.
Polynomial Expressions Involving Matrices
• If A is a square matrix, m x m, and if

is any polynomial, then we define

• Example:
Properties of the Transpose
• 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.
Invertibility of a Transpose
• Theorem: If A is an invertible matrix, then AT is also invertible and (AT)-1 = (A-1)T
• Example:
Elementary Matrices
• 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.
Elementary Matrices
• 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.
Elementary Matrices
• 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
Elementary Matrices
• 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.
Elementary Matrices
• Proof:

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

Let Ax=0 be the matrix form of the system

Elementary Matrices

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.
A Method for Inverting Matrices
• 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].
A Method for Inverting Matrices

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.

A Method for Inverting Matrices
• 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: