2
(6)
(7)
(8)
(9)
(10)
(11)
Lecture 2 (Row reduced Echelon form, elementary row operations)
Recall that the collection R of all the real numbers is not just a set. There are two binary
operations (namely, addition and multiplications) along with certain nice properties (namely,
commutativity, associativity, distributivity of multiplication over addition, also difference of
numbers and division by nonzero real numbers etc.)
Note that R can be replaced by C (complex numbers) or Q (rational numbers). However,
mostly we will use R. For solving a system of linear equations using N (natural numbers) or
Z (integers) only will not be considered. When we solve linear equations, we take difference
or/and ratio, so we always take some field (in this course, R, Q or C).
To have a geometric idea we consider R as a straight line and R2 as plane and R3 as the space
etc. If x̄ = (x1 , x2 , . . . , xn ), ȳ = (y1 , y2 , . . . , yn ) ∈ Rn and a ∈ R, define x̄ + ȳ = (x1 + y1 , x2 +
y2 , . . . , xn + yn ) and ax̄ = (ax1 , ax2 , . . . , axn ).
Introduce matrices and the symbol Mm×n (R). Recall properties of matrix addition and matrix
multiplication and recall trace and determinant of square matrices. Define column
� and
row matrix. If A is an n × n matrix with�(i, j)-th entry equal to ai,j then tr(A) = ni=1 aii
and for any j (1 ≤ j ≤ n), define det(A) = ni=1 aij Aij where Aij is (−1)i+j × det of the matrix
obtained from A by simply removing the i-th row and j-th column. It is not difficult to verify
using the definition that tr(A + B) = tr(A) + trB and tr(AB) = tr(BA). It may be harder to
verify that det(AB) = det A det B.
Define row reduced echelon matrix (RRE). A matrix is called row reduced echelon if the
following properties hold.
(a) Every zero row is below every nonzero row.
(b) The leading coefficient of every nonzero row is 1.
(c) A column which contains leading nonzero entry (which is 1) of a row has all other coefficients equal to zero.
(d) Suppose the matrix has r nonzero rows (and remaining m − r rows are zero). If leading
nonzero entry of i-th row (1 ≤ i 
≤ r) occurs
 in the ki -th column, then k1 < k2 < · · · < kr .
1 0 2
Consider the following matrices: a) 0 0 0is not RRE since it has a zero row preceding
0 1 0


1 0 1
a nonzero row, b) 0 2 0 is not RRE since, he second row is nonzero but its leading non
0 0 0


1 1 2
zero entry is 2 (�= 1), c) 0 1 1 is not RRE since the leading nonzero coefficient of the
0 0 0
second row is in the
second

column but the second column has another nonzero coefficien (in
1 0 2

the first row), d) 0 1 3 is not RRE since the leading nonzero coefficient of the third row
0 0 1


1 0 2
is in the third column and the third column has another nonzero coefficients , e) 0 1 3
0 0 0


0 1 2

is RRE, f) 1 0 3 is not RRE since in this there are two nonzero rows and k1 = 2 and
0 0 0
k2 = 1 violating K1 < k2 .
Recall: What is an equivalence relation (a relation which is reflexive, symmetric and transitive).
We are going to describe an equivalence relation (called row equivalence) on the collection
Mn (F) of n × n matrices with coefficients from F = R, C or Q.
Define elementary row operations. Namely,
(a) Multiplying the i-th row by a nonzero scalar λ denoted by Ri → λRi .
(b) Interchanging i-th row and j-th row denoted by Ri ↔ Rj .
(c) Replacing the i-th row by the sum of i-th row and µ multiple of j-th row denoted by
Ri → Ri + µRj .