Systems of Linear Equation and Matrices

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

2. 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

3. Introduction to Systems of Linear Equations

4. 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).

5. 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

6. 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:

7. 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

8. 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

9. 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.

10. 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.

11. Elementary Row Operations (Example) • r2= -2r1 + r2 • r3 = -3r1 + r3

12. Elementary Row Operations (Example) • r2 = ½ r2 • r3 = -3r2 + r3 • r3 = -2r3

13. Elementary Row Operations (Example) • r1 = r1 – r2 • r1 = -11/2 r3 + r1 • r2 = 7/2 r3 + r2 • Solution:

14. Gaussian Elimination

15. 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.

16. 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:

17. 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.

18. 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

19. 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

20. 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.

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

22. Matrices and Matrix Operations

23. Matrices and Matrix Operations

24. Inverses; Rules of Matrix Arithmetic

25. 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:

26. 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

27. 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.

28. 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.

29. 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

30. 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

31. 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.

32. 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:

33. 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

34. 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:

35. 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.

36. Powers of a Matrix • Example:

37. Polynomial Expressions Involving Matrices • If A is a square matrix, m x m, and if is any polynomial, then we define • Example:

38. 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.

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

40. Elementary Matrices and a Method for Finding A-1

41. 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.

42. 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.

43. 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

44. 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.

45. 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

46. 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.

47. 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].

48. 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.

49. 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:

50. Special Matrices: Diagonal Matrices, Triangular Matrices