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 Example 1 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 Example 2 Finding a Solution Set The equations are not linear. A solution of is a sequence of n numbers s1, s2, ..., sn Э 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.

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:

Properties of Matrix Operations 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 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. 0 for 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 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 Row operation on I that produces E Row operation on E that reproduces I Multiply row i by c ≠ 0 Multiply row i by 1/c Interchange rows i and j Add c times row i to row j Add –c times row i to row j

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