what is square matrix and elementary matrix??
explain with details,,
Answers
Step-by-step explanation:
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(R) when R is a field. They are also used in Gauss-Jordan elimination to further reduce the matrix to reduced row echelon form.
Answer:
Definition of Square Matrix: An n × n matrix is said to be a square matrix of order n. In other words when the number of rows and the number of columns in the matrix are equal then the matrix is called square matrix.
For example:
A=[ 2 7 9 1 4 2 8 6 3 ]
The number of rows of the above matrix = 3
The number of columns of the above matrix = 3
Since the number of rows and the number of columns are equal, the above matrix
Definition[Elementary matrices] Let n ≥ 1.
(1) Let 1 ≤ i, j ≤ n such that i 6= j, and let α ∈ R. We define Eij (α) to
be the n × n matrix obtained from the identity matrix In by changing the
entry in the position ij from 0 to α [note that we do allow α to be equal to
0].
(2) Let 1 ≤ i ≤ n and let α ∈ R, α 6= 0. We define Eii(α) to be the n × n
matrix obtained from the identity matrix In by changing the entry in the
position ii from 1 to α [note that we do allow α to be equal to 1].
(3) Let 1 ≤ i, j ≤ n such that i 6= j. We define Eij to be the the n × n
matrix obtained from the identity matrix In by interchanging the i-th and
j-th rows of In. Note that Eij = Eji.
We refer to Eij (α), Eii(α) and Eij in (1), (2) and (3) as elementary
matrices.
Example. For n = 3
E23(−9) =
1 0 0
0 1 −9
0 0 1
,
E33(7) =
1 0 0
0 1 0
0 0 7
and
E13 = E31 =
0 0 1
0 1 0
1 0 0