There were three types of elementary matrices covered in your
linear algebra course. Their purposes were to:

- Multiply a row by a scalar multiple,
- Swap two rows, and
- Add a scalar multiple of one row onto another.

Elementary matrices have nice properties: their inverses
are easy to calculate and their products with other matrices
are often easy to determine.

# Multiply a Row by a Scalar Multiple

Multiplying the *i*th row of an *n* × *n* matrix
by a scalar multiple *c* may done by multiplying on the left by the matrix which
is the identity matrix with entry *e*_{i,i} = *c*. For
example, multiplying the 3rd row of a 4 × 4 matrix may be done by
multiplying the matrix by the elementary matrix:

To visualize this in Matlab, run the following code:

M = [1 2 3 4 5;2 3 4 5 6;3 4 5 6 7;4 5 6 7 8;5 6 7 8 9]
E = eye( 5 ) % generate a 5x5 identity matrix (ha, ha)
E(2,2) = 1.271543 % set the 2,2 entry to 1.271543
E * M

If the row is being multiplied by a non-zero value, the
reciprocal of the elementary matrix is the identity
matrix with entry *e*_{i,i} = *1/c*.

# Swapping Two Rows

The elementary matrix which swaps two rows *i* and
*j* is the identity matrix with rows *i* and *j*
swapped.

To visualize this in Matlab, run the following code:

M = [1 2 3 4 5;2 3 4 5 6;3 4 5 6 7;4 5 6 7 8;5 6 7 8 9]
E = eye( 5 ) % generate a 5x5 identity matrix
E([2,5],:) = E([5,2],:) % swap rows 2 and 5 -- try to figure out how it works
E * M

The inverse of this elementary matrix is itself.

# Add a Scalar Multiple of One Row onto Another

The elementary matrix which adds to row *i*,
row *j* (≠ *i*) multiplied by *c* is the identity
matrix with entry *i,j* set to *c*. For example,
adding to row 3, 3.2 times row 1, is done by the following
elementary matrix:

To visualize this in Matlab, run the following code:

M = [1 2 3 4 5;2 3 4 5 6;3 4 5 6 7;4 5 6 7 8;5 6 7 8 9]
E = eye( 5 ) % generate a 5x5 identity matrix
E(4,2) = 0.1 % set entry (4,2) to 0.1
E * M % add to row 4, 0.1 times row 2

The inverse of this elementary matrix is the identity matrix
with entry *i,j* set to -*c*.