4.5. ELEMENTARY MATRICES 91

You use block multiplication(a b

c P

)(p q

r Q

)=

(ap+br aq+bQpc+Pr cq+PQ

)

Note that this all makes sense. For example, b = 1× n− 1 and r = n− 1× 1 so br is a1×1. Similar considerations apply to the other blocks.

Here is a very significant application. A matrix is called block diagonal if it has allzeros except for square blocks down the diagonal. That is, it is of the form

A =

A1 0

A2. . .

0 Am

where A j is a r j× r j matrix whose main diagonal lies on the main diagonal of A. Then byblock multiplication, if p ∈ N the positive integers,

Ap =

Ap

1 0Ap

2. . .

0 Apm

 (4.20)

Also, A−1 exists if and only if each block is invertible and in fact, A−1 is given by the abovewhen p =−1.

4.5 Elementary MatricesThe elementary matrices result from doing a row operation to the identity matrix. Recallthe following definition.

Definition 4.5.1 The row operations consist of the following

1. Switch two rows.

2. Multiply a row by a nonzero number.

3. Replace a row by a multiple of another row added to it.

The elementary matrices are given in the following definition.

Definition 4.5.2 The elementary matrices consist of those matrices which result by ap-plying a row operation to an identity matrix. Those which involve switching rows of theidentity are called permutation matrices1.

1More generally, a permutation matrix is a matrix which comes by permuting the rows of the identity matrix,which means possibly more than two rows are switched.