Summary - AP statistics Matrix algebra tutorial-Elementary Matrix Operations

Elementary Matrix Operations
There are three kinds of elementary matrix operations.


1. Interchange two rows (or columns).

2. Multiply each element in a row (or column) by a non-zero number.

3. Multiply a row (or column) by a non-zero number and add the result to another row (or column).


When these operations are performed on rows, they are called elementary row operations; and when they are performed on column
they are called elementary column operations.


Elementary Operation Notation
In many references, you will encounter a compact notation to describe elementary operations. That notation is shown below.


Operation description Notation

Row operations


1. Interchange rows i and j Ri <--> Rj


2. Multiply row i by s, where s ≠ 0 sRi --> Ri


3. Add s times row i to row j sRi + Rj --> Rj


Column operations


1. Interchange columns i and j Ci <--> Cj


2. Multiply column i by s, where s ≠ 0 sCi --> Ci


3. Add s times column i to column j sCi + Cj --> Cj



Elementary Operators

Each type of elementary operation may be performed by matrix multiplication, using square matrices called elementary operators.

For example, suppose you want to interchange rows 1 and 2 of Matrix A. To accomplish this, you could premultiply A by E to produce
as shown below.



0 1 1 3 5
R1 <--> R2 =
1 0 2 4 6

E A


0+2 0+4 0+6
R1 <--> R2 =
0+1 0+3 0+5