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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=-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:


    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:


    Gaussian Elimination


    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:


    Matrices and Matrix Operations


    Matrices and Matrix Operations


    Inverses; Rules of Matrix Arithmetic


    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

    (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

    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=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.


    Powers of a Matrix

    • Example:


    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 and a Method for Finding A-1


    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:


    Special Matrices: Diagonal Matrices, Triangular Matrices


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