Matrix multiplications that do row operations

Transcription

1 May 6, 204 Matrix multiplications that do row operations page Matrix multiplications that do row operations Introduction We have yet to justify our method for finding inverse matrices using row operations: that if row ops. there s a sequence of row operations [ A I] [ I B], then B = A. The use of the side-by-side matrix is really just a way to carry out an identical set of row operations on two different matrices, so what we really need to explain is: If there s a sequence of row operations that turns A into I and that turns I into B, why does B = A? An illuminating justification comes from learning a connection between row operations and matrix multiplication. The basic idea is that every row operation can be done by multiplying by an appropriate matrix. Finding a matrix for each row operation When we were solving linear systems using matrices, we used three kinds of row operations: (a) adding a multiple of one row to another row, (b) multiplying a row by a number, and (c) swapping two rows. For any such row operation, it s possible to find a matrix multiplication that does that row operation. Example: Find the matrix multiplication that does R + 3R2 R to any 2-by-3 matrix. Answer: 0 3 a b c a 3d b 3e c 3 f. d e f d e f. For each of the row operations listed below, find the 2-by-2 matrix that does that row operation. Check your answer by multiplying your chosen matrix with the general 2-by-3 a b c matrix. d e f a. 3R2 R2 b. 2 R R c. R2 4R R2

2 May 6, 204 Matrix multiplications that do row operations page 2 d. R R2 R e. Swap R R2. f. R2 R2 2. Write the 3-by-3 matrix that would do each of these row operations on a 3-by-4 matrix. [You do not have to write out the multiplications that confirm your answers, unless you are unsure and want to check whether you are right.] a. 2R3 R3 e. Swap R R3 b. Swap R2 R3 f. R3 5R R3 c. R2 + 4R R2 g. R R d. 4 R2 R2 h. R 2 R3 R

3 May 6, 204 Matrix multiplications that do row operations page 3 Summary: the elementary matrices An elementary matrix is a matrix that (when multiplied on the left) performs one of the basic row operations. All of the matrices found in problems and 2 are examples. Here is a summary of what the elementary matrices for each type of row operation will look like. type of row operation what the matrix must look like examples (3-by-3 s from problem 2) add a multiple of a row to almost the identity matrix, but with 0 0 another row a single 0 changed to another number R 3 5R R R 2 2 R 3 R multiply a row by a almost the identity matrix, but with 0 0 number a single changed to the number 2R 3 R swap two rows almost the identity matrix, but with 0 0 two pairs of s and 0 s reversed Swap R 2 R Why the inverse-finding procedure works Given: there exists a sequence of row operations that turns A into I and that turns I into B. To prove: that B = A. Proof: Call the elementary matrices that do the sequence of row operations: E, E2,, En. Based on what s given, we know that En E2 E A = I and En E2 E I = B Multiply the first equation on the right by A, and just simplify the second equation, to get En E2 E = A and En E2 E = B The left sides of these equations are identical, so transitivity gives that B = A. Reverse row operations and inverses of elementary matrices For every row operation, there s a row operation that reverses it. Whenever row operations reverse each other, the matrices that do the operations are inverses of each other. Example: R3 5R R3 can be reversed by doing R3 + 5R R Matrices 0 0 and 0 0 are inverses of each other Take En E2 E = A from the proof above. Multiply on the left by En then successively by each of the other elementary matrix inverses. This leads to I = A E E2 En. Finally a multiplication by A gives A = E E2 En. This means that we ve found a factorization of A as a product of elementary matrices.

4 May 6, 204 Matrix multiplications that do row operations page 4 Homework problems 3. Below are several row operations for matrices with 4 rows. For each: Write the 4-by-4 elementary matrix E that does this row operation. Find the row operation that reverses the given row operation, and write its elementary matrix. Check your answer by finding E on your calculator, which should be the matrix you wrote for the reverse row operation. a. 3R4 R4 b. R2 + 5R4 R2 c. R3 2R R3 d. Swap R2 R4 e. 2 5 R2 R2 f. R R4 R

5 May 6, 204 Matrix multiplications that do row operations page a. The inverse of matrix is the matrix 2 5. This inverse can be found using 2 3 row operations: row ops Following our standard procedure for choosing the row operations, there should be four row operation steps. Find these four row operations, and identify them below. Also record the matrix after each step. Identify your st row operation: Identify your 2nd row operation: Identify your 3rd row operation: Identify your 4th row operation: b. Write down the elementary matrix corresponding to each of the four row operations. E = E2 = E3 = E4 = c. Compute the matrix multiplication E4 E3 E2 E (do it by hand: first multiply E4 E3, then multiply by E2, then multiply by E. (If you do everything correctly, the final answer should be A.) d. Write the inverses of the elementary matrices from above. (The easiest way to find them may be to first identify the row operation that reverses each row operation.) E = E2 = E3 = E4 = e. Compute the matrix multiplication E E2 E3 E4 to verify that it equals A. If so, you have just found a factorization of A.

6 May 6, 204 Matrix multiplications that do row operations page Let A = 0. Repeating the methods of problem 4, calculate A, and find a factorization of A as a product of elementary matrices. 6. Using the inverse matrix found in problem 5, solve the linear system x + 5z = 2 y z = 0 3x + 4y + 2z = 8 Hint: Write the system in the form AX = C where X = x. Then AX = C X = A C. y z