Section 2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems

Transcription

1 Section. Systems of Linear Equations: Underdetermined and Overdetermined Systems Infinitely Many Solutions: If an augmented coe cient matrix is in row-reduced form and there is at least one row which consists entirely of zeros, then,in most cases, thesystemhasinfinitely many solutions and we use parameter t and/or s to write the solution. Note: The case when this assumption is not always true is when the system is overdetermined. Example 1: Solve the following system of equations x +y z = x y z =1 x +y 5z = No solution If an augmented coe cient matrix is in row-reduced form and there is at least one row which consists entirely of zeros to the left of the vertical line and a nonzero entry to the right of the line (the very last entry on that row), then the system has no solution. Example : Solve the following system of equations x + y + z =1 x y z =4 x +5y +5z = 1

2 Underdetermined System A system is underdetermined if there are than there are variables. Note: An underdetermined system can have no solution or infinitely many solutions. equations Example : Solve the following system of equations x +4y 18z =44 4x y +4z = 44 Overdetermined System A system is overdetermined if there are equations than there are variables. Note: An overdetermined system can have a unique solution, no solution or infinitely many solutions. Example 4: Solve the following system of equations 14x +y = 10 0x 4y =0 6x +6y = 0 Fall 017, Maya Johnson

3 Example 5: Solve the following systems of equations. (If there are infinitely many solutions, enter a parametric solution using t and/or s). a) y +z =1 x y z =4 x +y z =5 b) x +y +z =10 8x +8y +8z = 4x +5y +z = k ;ED k!:#'n.sow# Fall 017, Maya Johnson

4 c) x y + z = x +y 4z = 7 x y +5z =8 x 8y +9z =17 d) f. :I Ef*Ee n# does not mean many infinitely x y +4z = x + y z = 1 x +4y 8z = 5 solutions 4 Fall 017, Maya Johnson

5 Example 6: Adietitianwishestoplanamealaroundthreefoods. Themealistoinclude16, 80 units of vitamin A, 5, 090 units of vitamin C, and 1, 050 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the table below. Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements. Food I Food II Food III vitamin A vitamin B Calcium p. ;em e. i o t D 5 Fall 017, Maya Johnson

6 Section.4 Matrices What is a Matrix? A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has size m n. Theentryintheith row and jth column of a matrix A is denoted by a ij. Note: If A is an n n matrix, then we say A is a square matrix. Example 1: Given the matrix A = a) what is the size of A? b) find a 14, a 1, a 1,anda 4 Equality of Matrices Matrices A and B are equal if and only if they have the same size and they have the same corresponding entries (i.e. a ij = b ij for all values of i and j). Example : Are the two matrices below equal? A = B = 4 (5 1) 8 8 (+) (1 11) Example : If we know the matrices below are equal, find x, y, andz. 6 4 x 9 5 y 10 z = y 10 z Fall 017, Maya Johnson

7 Adding and Subtracting Matrices If matrices A and B have the same size (both m n matrices) then: 1. The sum A + B is obtained by adding the corresponding entries in both matrices (a ij + b ij for all values of i and j) andtheresultingmatrixisstillanm n matrix.. The di erence A B is obtained by subtracting the corresponding entries in both matrices (a ij b ij for all values of i and j) andtheresultingmatrixisstillanm n matrix. Note: You CANNOT add or subtract two matrices that have di erent sizes. Also, A + B = B + A BUT A B 6= B A. Scalar Product If c is a real number and A is an m n matrix, then the scalar product ca is obtained by multiplying every entry in A by c (ca ij for all values of i and j) and the resulting matrix is still an m n matrix. Example 4: Perform the indicated operations Transpose of a Matrix The transpose of a matrix A, denoteda T,isobtainedbyinterchangethe rows and the columns of A. Therefore, if A is an m n matrix with entries a ij then A T is an n m matrix with entries a ji. Example 5: Find the transpose of the matrix. " # 7 Fall 017, Maya Johnson

8 Example 6: Matrix L is a 4 7matrix,matrixM is a 7 7matrix,matrixN is a 4 4matrix,and matrix P is a 7 4 matrix. Find the dimensions of the sums below, if they exist. (If an answer does not exist, write DNE.) a) L + M b) L + P T c) M + N d) N + N Example 7: Find the values of a, b, c, andd in the matrix equation below. " a c b d # + " 4 5 # T = " # 8 Fall 017, Maya Johnson

9 Example 8: The Campus Bookstore s inventory of books is as follows. Hardcover: textbooks, 5119; fiction, 1948; nonfiction, 4; reference, 1514 Paperback: fiction, 57; nonfiction, 157; reference, ; textbooks, 1849 The College Bookstore s inventory of books is as follows. Hardcover: textbooks, 698; fiction, 054; nonfiction, 1986; reference, 189 Paperback: fiction, 0; nonfiction, 1719; reference, 850; textbooks, 477 a) Represent the Campus s inventory as a matrix A. b) Represent the College s inventory as a matrix B. c) The two companies decide to merge, so now write a matrix C that represents the total inventory of the newly amalgamated company. t.int isit it :I A : :S ii: :D = He is% :Y : :D 9 Fall 017, Maya Johnson