We introduce elementary matrices which have the following property: multiplying A on the left by an elementary matrix has the effect of performing a row operation to A.
Let i, j be integers between 1 and n inclusive, and assume i is not equal to j. Let lambda be a real number. Then E_{i j}(lambda) is the matrix which has:
1s on the diagonal,
0s everywhere else except in the i jth position, where there is a lambda.
Such a matrix is an elementary matrix of type I.
For example, if n = 3 you get things like: E_{1 2}(4) = 1 , 4 , 0 ; 0 , 1 , 0 ; 0 , 0 , 1 and E_{32}(minus 5) = 1 , 0 , 0 ; 0 , 1 , 0 ; 0 , minus 5 , 1. If n = 4, you get things like E_{2 3}(7) = 1 , 0 , 0 , 0 ; 0 , 1 , 7 , 0 ; 0 , 0 , 1 , 0 ; 0 , 0 , 0 , 1.
Let A and _{i j}(lambda) be n-by-n matrices. Then E_{i j}(lambda) times A is the matrix obtained from A by doing R_i maps to R_i + lambda R_j.
Let's consider the case i less than j (the other case is similar so we omit it). Consider the product E_{i j}(lambda) times A The only difference the lambda makes is when we multiply the ith row into a column of A (say the kth column). Instead of just picking up 1 times A_{i k}, we get 1 times A_{i k} plus lambda times A_{j k}.
In other words, the result E_{i j}(lambda) times A is obtained from A by adding lambda times row j to row i.
Let i be an integer between 1 and n inclusive and let lambda not equal to zero be a real number. Define E_i(lambda) to be the matrix which has 1s on the diagonal everywhere except in the i ith position, where it has a lambda. Such a matrix is an elementary matrix of type II.
If n = 3, E_2(lambda) = 1 , 0 , 0 ; 0 , \lambda , 0 ; 0 , 0 , 1 and E_3(minus 1) = 1 , 0 , 0 ; 0 , 1 , 0 ; 0 , 0 , minus 1.
E_i(lambda) times A is obtained from A by performing the row operation R_i maps to lambda R_i.
In the product, E_i(lambda) times A most of the entries are multiplied by 1, except those in the ith row of A, which are rescaled by lambda.