20 CHAPTER 1 Systems of Linear Equations REPLACE ONE ROW BY ADDING THE SCALAR MULTIPLE OF ANOTHER ROW The last type of operation is slightly more complicated. Suppose that we want to write down the elementary matrix that corresponds to the operation αr i + β R j R j where R i is row i and R j is row j. To do this, we start with I n and modify row j in the following way: Replace the element in column i by α. Replace the element in column j by β. EXAMPLE 1-8 Fora3 3 matrix A, write down the three elementary matrices that correspond to the row operations R 2 R 3 4R 2 R 2 3R 1 + R 3 R 3 SOLUTION 1-8 We start with I 3 I 3 The row operation R 2 R 3 is represented by swapping rows 2 and 3 in the I 3 matrix 0 1 0 To represent 4R 2 R 2, we replace the second row of I 3 with 0 4 0
CHAPTER 1 Systems of Linear Equations 21 Now we consider 3R 1 + R 3 R 3. We will modify row 3, which is the destination row, in the I 3 matrix. We will need to replace the element in the first column, which is a 0, with a 3. The element in the third column is unchanged because the scalar multiple is 1, and so we use EXAMPLE 1-9 Represent the operations 3 0 1 2R 1 R 2 R 2 and 4R 2 + 6R 3 R 3 with elementary matrices in a 3 3 system. SOLUTION 1-9 To represent 2R 1 R 2 R 2, we will modify row 2 of I 3. We replace the element in the first column with a 2, and change the element in the second column with a 1. This gives 2 1 0 To represent the second operation, we replace the third row of I 3. The operation is 4R 2 + 6R 3 R 3, and so we replace the element at the second column with a 4, and the element in the third column with a 6, which results in the matrix 0 4 6 EXAMPLE 1-10 Fora4 4 matrix, find the elementary matrix that represents 2R 2 + 5R 4 R 4
22 CHAPTER 1 Systems of Linear Equations SOLUTION 1-10 To construct an elementary matrix, we begin with a matrix with 1s along the diagonal and 0s everywhere else. For a 4 4 matrix, we use 1 0 0 0 0 1 0 0 I 4 0 0 The destination row is the fourth row, and so we will modify the fourth row of I 4. The operation involves adding 2 times the second row to 5 times the fourth row. And so we will replace the element located in the second column by 2 and the element in the fourth column by 5, which gives 1 0 0 0 0 1 0 0 0 0 2 0 5 Implementing Row Operations with Elementary Matrices Row operations are implemented with elementary matrices using matrix multiplication. We will explore matrix multiplication in detail in the next chapter, but it turns out that matrix multiplication using an elementary matrix is particularly simple. For now, we will show how to do this for 2 2 and 3 3 matrices. MATRIX MULTIPLICATION BY A 2 2 ELEMENTARY MATRIX Let E be an elementary matrix and A be an arbitrary 2 2 matrix given by a b A c d
CHAPTER 1 Systems of Linear Equations 23 We have two cases to consider, operations on the first and second rows. An arbitrary operation on the first row is represented by E 1 α β 0 1 The product E 1 A is given by α β a b αa + βc αb + βd E 1 A 0 1 c d c d An operation on row 2 is given by and the product E 2 A is 1 0 E 2 α β [ 1 0 a b a E 2 A α β c d αa + βc b αb + βd EXAMPLE 1-11 Consider the matrix A 2 5 4 11 Implement the row operations 2R 1 R 1 and 3R 1 + R 2 R 2 using elementary matrices. SOLUTION 1-11 The operation 2R 1 R 1 is represented by the elementary matrix E 2 0 0 1
24 CHAPTER 1 Systems of Linear Equations Using the formulas developed above, we have 2 0 2 5 (2)( 2) + (0)(4) (2)(5) + (0)(11) EA 0 1 4 11 4 11 4 + 0 10+ 0 4 10 4 11 4 11 The elementary matrix that represents 3R 1 + R 2 R 2 is found to be 1 0 E 3 1 The product is [ 1 0 2 5 2 5 EA 3 1 4 11 ( 3)( 2) + (1)(4) ( 3)(5) + (1)(11) 2 5 2 5 6 + 4 15 + 11 10 4 ROW OPERATIONS ON A 3 3 MATRIX Row operations on 3 3 matrix A are best shown with example. The multiplication techniques are similar to those used above. EXAMPLE 1-12 Consider the matrix A Implement the row operations 2R 2 R 2, R 1 R 3, 4R 1 + R 2 R 2 using elementary matrices. SOLUTION 1-12 The elementary matrix that corresponds to 2R 2 R 2 is given by E 1 0 2 0
CHAPTER 1 Systems of Linear Equations 25 The operation is implemented by computing the product of this matrix with A: E 1 A 0 2 0 [ (0)(7) + (2)(0) + (0)( 2) (0)(7) + (2)(1) + (0)( 2) (0)(7) + (2)(4) + (0)( 2) 0 2 8 The swap operation R 1 R 3 can be implemented with the matrix E 2 1 0 0 In this case rows 1 and 3 have been changed. So we will multiply both rows in this case. The result is E 2 A 1 0 0 [ (0)(7) + (0)(0) + (1)( 2) (0)( 2) + (0)(1) + (1)(3) (0)(3) + (0)(4) + (1)(5) 0 1 4 (1)(7) + (0)(0) + (0)( 2) (1)( 2) + (0)(1) + (0)(3) (1)(3) + (0)(4) + (0)(5) 0 + 0 2 0+ 0 + 3 0+ 0 + 5 0 1 4 7 + 0 + 0 2 + 0 + 0 3+ 0 + 0
26 CHAPTER 1 Systems of Linear Equations Finally, we can implement the operation 4R 1 + R 2 R 2, using the elementary matrix E 3 4 1 0 We find E 3 A 4 1 0 ( 4)(7) + (1)(0) ( 4)( 2) + (1)(1) ( 4)(3) + (1)(4) 28 + 0 8+ 1 12 + 4 28 9 8 Homogeneous Systems A homogeneous system is a linear system with all zeros on the right-hand side. In general, it is a system of the form a 11 x 1 + a 12 x 2 + +a 1n x n 0 a 21 x 1 + a 22 x 2 + +a 2n x n 0. a m1 x 1 + a m2 x 2 + +a mn x n 0 When a system is put in echelon form, if the system has more unknowns than equations, then it has a nonzero solution. A system in echelon form with n equations and n unknowns has only the zero solution, meaning that only (x, y, z) (0, 0, 0) solves the system.