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Example:Finding Inverse of a Matrix (Elementary Transformation)

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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)




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).


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)


§  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)


       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].




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




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.