Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University
Standard of Competency: Understanding the properties of systems of linear equations, matrices, and their use in problem solving
Basic Competencies: 3
A. Introduction to Systems of Linear Equations Definition We define a linear equation in the n variables x, x,, x n to be one that can be expressed in the form a x + a x + + a n x n = b where a, a,, a n and b are real constants. The variables in a linear equation are sometimes called the unknowns.
Examples :
Definition A solution of a linear equation a x + a x + + a n x n = b is a sequence of n numbers s, s,, s n such that the equation is satisfied when we substitute x = s, x = s,, x n = s n. The set of all solutions of the equation is called its solution set or the general solution of the equation
Examples : Find the solution set of (a) and (b) Solutions : (a) (b)
Linear equations Systems A finite set of linear equations in the variables x, x,, x n is called a system of linear equations or a linear systems. A sequence of n numbers s, s,, s n is called a solution of the linear system if x = s, x = s,, x n = s n is a solution of every equation in the linear system.
For example, the system has the solution However, is not the solution
Every system of linear equations has no solutions, or has exactly one solution, or has infinitely many solutions. The lines l and l may be parallel, in which case there is no intersection and consequently no solution to the system. The lines l and l may intersect at only one point, in which case the system has exactly one solution. The lines l and l may coincide, in which case there are infinitely many points of intersection and consequently infinitely many solutions to the system.
The General Form of Linear Systems An arbitrary system of m linear equations in n variables can be written as a x + a x + + a n x n = b a x + a x + + a n x n = b : : : : a m x + a m x + + a mn x n = b m where x, x,, x n are the unknowns and the subscripted a s and b s denote real constants.
Augmented Matrices A system of m linear equations in n unknowns can be written as follows: m mn m m n n b a a a b a a a b a a a L M M O M M L L This is called the augmented matrix for the system.
For example, the augmented matrix for the system of equations
Elementary Row Operations Three types of operations to eliminate unknowns:. Multiply an equation through by a nonzero constant.. Interchange two equations. 3. Add a multiple of one equation to another. Three types of operations on the rows of the augmented matrix:. Multiply a row through by a nonzero constant.. Interchange two rows. 3. Add a multiple of one row to another row.
Example In the following column below we solve a system of linear equations by operating on the equations in the systems In the following column below we solve a system of linear equations by operating on the rows of the augmented matrix x + y + z = 9 x + 4y 3z = 3x + 6y 5z = 3 4 6 3 5 9
Add - times the first equation to the second equation and add -3 times the first equation to the third equation to obtain Add - times the first row to the second row and add -3 times the first row to the third row to obtain x + y + z = 9 y 7z = -7 3y z = -7 3 7 9 7 7
Multiply the second equation by ½ to obtain x + y + z = 9 y 7/ z = - 7/ 3y z = -7 Add -3 times the second equation to the third to obtain x + y + z = 9 y 7/ z = - 7/ -- ½ z = -3/ Multiply the second row by ½ to obtain 3 7 9 7 7 Add -3 times the second row to the third to obtain 7 9 7 3
Multiply the third equation by - to obtain x + y + z = 9 y 7/ z = - 7/ z = 3 Add - times the second equation to the first to obtain Multiply the third row by- to obtain 7 9 7 Add - times the second row to the first to obtain 3 x + / z = 35/ y 7/ z = - 7/ z = 3 7 35 7 3
Add -/ time the third equation to the first and 7/ time the third equation to the second to obtain Add -/ time the third row to the first and 7/ time the third row to the second to obtain x = y = z = 3 3 Thus the solution of the system of linear equations is x =, y =, z = 3.
Gaussian Elimination
Reduced Row-Echelon Form A matrix is said to be in reduced row-echelon form, if the following properties are satisfied:. If a row does not consist entirely of zeros, then the first nonzero number in the row is a. (we call this a leading ).. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix. 3. In any two successive rows that do not consist entirely of zeros, the leading in the lower row occurs farther to the right than the leading in the higher row. 4. Each column that consists a leading has zeros everywhere else. A matrix having properties, and 3 is said to be in row-echelon form. (Thus, a matrix in reduced row-echelon form is of necessity in row-echelon form, but not conversely.)
Homogeneous Linear Systems A system of linear equations is said to be homogeneous if the constant terms are all zero, the system has the form a x + a x + + a n x n = a x + a x + + a n x n = : : : : a m x + a m x + + a mn x n = Every homogeneous system of linear equations is consistent, since all such systems have x =, x =,, x n = as a solution. This solution is called the trivial solution, if there are other solutions, they are called nontrivial solution.
Example Solve the following homogeneous system of linear equations by Gauss-Jordan eliminations. x + x - x 3 + x 5 = -x - x + x 3 3x 4 + x 5 = x + x - x 3 - x 5 = x 3 + x 4 + x 5 =
Solution: The augmented matrix for the system is 3 Reducing this matrix to reduced row-echelon form, we obtain
The corresponding system of equations is x + x + x 5 = x 3 + x 5 = x 4 = Thus the general solution is x = -s t, x = s, x 3 = -t, x 4 =, x 5 = t, where s and t are parameters. Note: The trivial solution is obtained when s = t =.
Theorem A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions.
Problem Solve the following system of nonlinear equations for x, y and z. Problem x + y + z = 6 x -y + z = x + y -z = 3 Show that the following nonlinear system has eighteen solutions if α π, β π, γ π. sin α + cos β + 3 tan γ = sin α + 5 cos β + 3 tan γ = -sin α - 5 cos β + 5 tan γ =
Matrices and matrix Operations By: Tri Atmojo K and Mardiyana Mathematics Education Sebelas Maret University
In the previous Section we used rectangular arrays of numbers, called augmented matrices, to abbreviate systems of linear equations. However, rectangular arrays of numbers occur in other contexts as well. For example, the following rectangular array with three rows and seven columns might describe the number of hours that a student spent studying three subjects during a certain week:
If we suppress the headings, then we are left with the following rectangular array of numbers with three rows and seven columns, called a matrix :
Definition A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix. Definition Two matrices are defined to be equal if the have the same size and their corresponding entries are equal. Definition If A and B are matrices of the same size, then the sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A.
Definition If A is any matrix and c is any scalar, then the product ca is the matrix obtained by multiplying each entry of A by c. Definition If A is an m x r matrix and B is an r x n matrix, then the product AB is the m x n matrix whose entries are determined as follows. To find the entry in row i and column j of AB, single out row I from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together and then add up the resulting products.
For example, if A, B, and C are the matrices in the earlier Example, then
Definition If A is any m x n matrix, then the transpose of A, denoted by A T, is defined to be the n x m matrix that results from interchanging the rows and columns of A, that is, the first column of A T is the first row of A, the second column of A T is the second row of A, and so forth. Definition If A is a square matrix, then the trace of A, denoted by tr(a), is defined to be the sum of the entries on the main diagonal of A. The trace of A is undefined if A is not a square matrix.
Definition If A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, where I is an identity matrix, then A is said to be invertible and B is called an inverse of A. Theorem 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) - = B - A -.
Theorem If A is an invertible matrix, then a). A - is invertible and (A - ) - = A b). A n is invertible and (A n ) - = (A - ) n for n =,, c). For any nonzero scalar k, the matrix ka is invertible and (ka) - = (/k) A -. d). A T is invertible and (A T ) - = (A - ) T. Proof: Exercise for student.
Problem A square matrix A is called symmetric if A T = A and skewsymmetric if A T = -A. Show that if B is a square matrix, then a). BB T and B + B T are symmetric. b). B B T is skew-symmetric. Problem Let A be a square matrix. a). Show that (I A) - = I + A + A + A 3 + A 4 if A 5 =. b). Show that (I A) - = I + A + A + + A n if A n+ =.
Elementary Matrices and a Method for Finding A - Definition An n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix I n by performing a single elementary row operation. Theorem If the elementary matrix A results from performing a certain row operation on I m and if A an m x n matrix, then the product EA is the matrix that results when this same row operation is performed on A.
SOME EXAMPLES OF ELEMENTARY MATRICES
EXAMPLE :
Theorem Every elementary matrix is invertible, and the inverse is also an elementary matrix. Theorem If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false. a). A is invertible. b). Ax = has only the trivial solution. c). The reduced row-echelon form of A is I n. d). A is expressible as a product of elementary matrices.
Remark: To find the inverse of an invertible matrix A, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on I n to obtain A -.
Example: Find the inverse of A = 5 3 3 8
Solution :
Thus,