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There are three types of elementary matrices, each corresponding to one of the types of elementary row operations. Note that elementary matrices are necessarily ...
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Math 571
Consider the following situation: A is a matrix, possible augmented, and U is the reduced row echelon form
of A. The U is obtained from A by a series of elementary row operations. However, these operations are, in
some sense, external to the matrix A. It turns out that we can accomplish this row reduction by multiplying
A by a sequence of matrices Ei called elementary matrices. In other words, U = Ek · · · E 1 A. There are three
types of elementary matrices, each corresponding to one of the types of elementary row operations. Note that
elementary matrices are necessarily n × n.
We will assume that all of our matrices are of size n × n and our notation will not refer to their size. Also all
specific examples will be 3 × 3 matrices. For convenience let
a b c
d e f
g h i
Notation: Pij , i 6 = j
Definition: Pij = the n × n identity matrix In with row i and row j interchanged
Examples: P 12 =
d e f
a b c
g h i
b a c
e d f
h g i
Multiplication on Left: Interchanges row i and row j of A.
Multiplication on Right: Interchanges column i and column j of A.
Inverse: P
− 1 ij = Pij
Determinant: det(Pij ) = − 1
Notation: Di(t), t 6 = 0
Definition: Di(t) = the n × n identity matrix In with t in row i, column i
Examples: D 3 (5) =
a b c
d e f
5 g 5 h 5 i
a b 5 c
d e 5 f
g h 5 i
Multiplication on Left: Multiplies row i of A by t.
Multiplication on Right: Multiplies column i of A by t
Inverse: Di(t)
− 1 = Di(1/t)
Determinant: det(Di(t)) = t
Notation: Eij (t), i 6 = j
Definition: Eij (t) = the n × n identity matrix In with t in row i, column j
Examples: E 13 (4) =
a + 4g b + 4h c + 4i
d e f
g h i
a b 4 a + c
d e 4 d + f
g h 4 g + i
Multiplication on Left: Adds t times row j of A to row i of A.
Multiplication on Right: Adds t times column i of A to column j of A.
Inverse: Eij (t) − 1 = Eij (−t)
Determinant: det(Eij (t)) = 1