Explanation:
Matrix
multiplication
Schematic
depiction of the matrix product AB of two matrices A and B.
Multiplication of two matrices is defined
only if the number of columns of the left matrix is the same as the number of
rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are
given by dot product of the corresponding row of A and the corresponding column
of B:
where 1
≤ i ≤ m and 1 ≤ j ≤ p.
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Matrix_(mathematics)
Elementary matrix
In mathematics, an elementary matrix is a simple matrix which differs from the identity matrix by one single elementary row
operation. The elementary matrices generate the general linear group of invertible matrices. Left
multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication
(post-multiplication) represents elementary column operations.
Use in
solving systems of equations
Elementary row operations do not change the
solution set of the system of linear equations represented by a matrix, and
are used in Gaussian elimination (respectively, Gauss-Jordan elimination) to
reduce a matrix to row echelon form (respectively, reduced row echelon form).
Operations
There are three types of elementary matrices, which correspond to
three types of row operations (respectively, column operations):
Row
switching
A row
within the matrix can be switched with another row.
Row
multiplication
Each
element in a row can be multiplied by a non-zero constant.
Row
addition
A row
can be replaced by the sum of that row and a multiple of another row.
(Our solved example in mathguru.com uses
this concept)
Row-multiplying
transformations
This transformation, Ti(m),
multiplies all elements on row i by m where m is non zero. The matrix resulting in
this transformation is obtained by multiplying all elements of row i of the identity matrix by m.
(Our solved example in mathguru.com uses
this concept)
Properties
§ The
inverse of this matrix is: Ti(m)−1 = Ti(1/m).
§
The matrix and its inverse are diagonal
matrices.
§
det [Ti(m)] = m.Therefore for a conformable square matrix A: det[Ti(m)A] = m det[A].
Row-addition transformations
This transformation, Tij(m),
adds row j multiplied by m to row i. The matrix resulting in this
transformation is obtained by taking row j of the identity matrix, and adding it m times to row i.
These are also called shear
mappings or transvections
(Our solved example in mathguru.com uses
this concept)
Properties
•
Tij(m)−1 = Tij(−m) (inverse matrix).
•
The matrix and its inverse are triangular
matrices..
•
det[Tij(m)]
= 1. Therefore, for a conformable square matrix A: det[Tij(1)A]
= det[A].
http://en.wikipedia.org/wiki/Elementary_matrix
Invertible matrix
In linear algebra an n-by-n (square) matrix A is called invertible or nonsingular or non degenerate, if there exists an n-by-n matrix B such that
where In denotes the n-by-n identity matrix and the multiplication used
is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A−1. (Our solved example in mathguru.com uses this concept)
It follows
from the theory of matrices that if
for finite square matrices A and B,
then also
http://en.wikipedia.org/wiki/Invertible_matrix
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