Elementary Matrices
An elementary matrix is a certain simple type of invertible matrix that differs only slightly from the
identity matrix. There are three generic types of elementary matrices. We describe these and list all the 3
x 3 elementary matrices.
Elementary Matrices of Type I (Permutation matrices). These are simply the identity matrix with the
columns interchanged.
Example. The 3 × 3 type I elementary matrices are the following
:
100
100
010
001
001
010
010 , 001 , 100 , 100 , 010 ,
000 .
001
101
010
001
010
100
Elementary Matrices of Type II. These are simply the identity matrix with one of the diagonal entries
replaced by a nonzero constant a.
Example. The 3 × 3 type II elementary matrices are the following:
a00
100
100
010 , 0a0 , 010
001
001
00a
Elementary Matrices of Type III (Shear Matrices). These are simply the identity matrix with at most one
nonzero number a in an off diagonal position.
Example. The 3 × 3 type III elementary matrices are the following:
1a0
10a
100
100
100
100
010 , 010 , a10 , 01a , 010 , 010
001
001
001
001
a01
0a1
for a ! 0.
Inverses of Elementary Matrices
All the elementary matrices are invertible. In fact, the inverses of elementary matrices can be
obtained by inspection.
Inverse of a Type I Elementary Matrix. The inverse of such a matrix is simply its transpose.
We can verify this using the dot product form of matrix multiplication. Here an elementary matrix
T
of type I has the form [ei(1), … , ei(n)] , where {i(1), … , i(n)} = {1, … ,n}. Here the vectors ej are the
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column vectors
T
T
e1 = (1, 0, … , 0) , … , en = (0, 0, … , 1) .
So the matrix product is
T
[ei(1), … , ei(n)] [ei(1), … , ei(n)] = (ei(s)"ei(t))s,t = (# st)s,t = I.
where #s,t is the Kronecker delta given by
#s , t =
1 if s = t .
0 if s ! t
010
001
Example. The inverse of
0 0 1 is 1 0 0 .
100
010
Inverse of a Type II Elementary Matrix. If the type II matrix is the identity matrix except for a ! 0 in the
i,i term, then the inverse of the matrix is the identity matrix except for 1/a in the i,i term.
Example. The inverse of
100
1
0 a 0 is
0 1/a 0 .
001
0
0
0
0
1
Inverse of a Type III Elementary Matrix. If the type III matrix is the identity matrix except for a ! 0 in the
i,j term for i ! j, then the inverse of the matrix is the identity matrix except for - a in the i,j term.
Example.
1a0
010
001
-1
1 -a 0
=
0 1 0 .
0 0 1
Multiplication on the Left by Elementary Matrices
Multiplication of an n x p matrix A on the left by an elementary matrix E causes certain changes in
the rows of the matrix A. We describe the specific changes below.
Multiplication on the left by an elementary matrix E of type I. Let E by the elementary matrix
E =[e1, e2, … , ei-1, ej, ei+1, … , ej-1, ei, ej+1, … en]
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T
(the identity matrix with the i and j column interchanged); then EA is the matrix A except that the i and j
row of A have been interchanged.
Example.
a11 a12 a13 a14
100
a11 a12 a13 a14
001
a21 a22 a23 a24 = a31 a32 a33 a34 .
010
a31 a32 a33 a34
a21 a22 a23 a24
Other examples are similar but depend on the introduction of the concept of permutation. We defer this
to later when we discuss determinants.
Multiplication on the left by an elementary matrix E of type II. Let E by the elementary matrix equal to
the identity matrix except for a ! 0 in the i,i place. Then EA is equal to the matrix A except that the ith row
of A has been multiplied by A.
Example.
a11 a12 a13 a14
100
a11 a12 a13 a14
0a0
a21 a22 a23 a24 = aa21 aa22 aa23 aa24 .
001
a31 a32 a33 a34
a31 a32 a33 a34
Multiplication on the left by an elementary matrix E of type III. Let E by the elementary matrix equal to
the identity matrix except for a in the i,j place where i ! j. Then EA is equal to the matrix A except that A
has a new jth row equal to the jth row of A plus a times the ith row of A.
Example.
a11 +aa21 a12 + aa22 a13 +aa23 a14 + aa24
1a0
a11 a12 a13 a14
010
a21 a22 a23 a24 =
a21
a22
a23
a24
001
a31 a32 a33 a34
a31
a32
a33
a34
.
Here a is in the 1,2 place and so there is a new first row equal to the original first row of A added to a
times the second row of A.
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