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Elementary Matrices

An n×n matrix is called an elementary matrix if it can be obtained from the n×n identity matrix In by performing a single elementary row operation.

Properties of Elementary Matrices:
a.  If the elementary matrix
E results from performing a certain row operation on Im  and if A is an m
×n matrix, then the product EA is the matrix that results when this same row operation is performed on A.
b.  Every elementary matrix is invertible, and the inverse is also an elementary matrix.


Example 1: Give four elementary matrices and the operations that produce them.
Solution: Listed below are four elementary matrices attached with the operations that produced them.
a. 
Multiply the second row of I2 by -3.
b.  Interchange the second and fourth rows of
I4.
c.  Add 3 times the third row of I3 to the first row.
d. 
Multiply the first row of I3 by 1.

Example 2: Verify first property of elementary matrices for the following 3×4 matrix.
Solution: Consider the matrix

and consider the elementary matrix


which results from adding 3 times the first row of I3 to the third row. The product EA is


which is precisely the same matrix that results when we add 3 times the first row of A to the third row. Hence the property is verified.

Example 3: What should we pre-multiply  by if we want to multiply row 3 by m?
Solution: We start with the 4×4 identity matrix , we then multiply row three by m to obtain .This is the desired elementary matrix. We can check that if we multiply A by this matrix, the resulting matrix will be A in which row three has been multiplied by m.



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