2. Linear Equations Objectives: 1.Introduction to Gaussian Elimination. 2. Using multiple row operations. 3. Exercise - let’s see if you can do it. Refs: B & Z 4.2, 4.3.

Example 1 (revisited): constantxy Use operation (A) Use operation (C) Use operation (B) ~ ~

~ ~ Use operation (C) Bingo! Read off solution. xyconstant Solution: x=1 y=5

We have just used a procedure known as Gaussian elimination (or row reduction) which transforms a matrix The procedure also applies to larger matrices. into

Example 2: Solve for x, y and z: The first step is to construct the augmented matrix: x coefficients constant terms z coefficients y coefficients

Our aim is to produce an equivalent augmented matrix which has 1’s on the diagonal and zeroes elsewhere (on the LHS). ~ ~ 2 Use (A) to get a 1 in the top left corner Use (C) to get a 0 in the position indicated Use (C) to get a 0 in the bottom left position

Use (C) to get a 0 in the position indicated Use (C) to get a 0 in the position indicated Use (B) to get a 1 in the bottom right corner ~ -2 ~ 4 Use (B) to get a 1 in the centre -2 ~5~

Solution: Always substitute these values back into ALL of your equations to check your solution. Note: It is very easy to make algebraic mistakes!!! -2 ~ ~ Use (C) to get a 0 in the position indicated We now have the required form xconstantzy ~ Use (C) to get a 0 in the Top right corner

Check:

Using multiple operations We can alter more than one row at a time to speed up the Gaussian elimination procedure. Example 3: ~ ~ ~ No problem - we have saved some time. (Here we are using multiple (B) Operations)

Example 4: We can perform multiple (C ) operations provided at least one row is kept constant and only multiples of it are used to perform the other operations. ~~

Obviously, performing multiple (A) type operations causes no problem. Exercise 1: Solve the following system of simultaneous equations: Example 4 (continued): ~ No problem - We kept R 1 constant And used it to get R 2 and R 3

Solution to Exercise ~ ~ ~ ~ ~

xyz (x,y,z)=(3,-2,1)

You can now attempt Q1 and Q2 from Exercise Sheet 1 of the Orange Book