Elementary transformations of matrices
Elementary transformations of a matrix find a wide application in various mathematical problems.
For example, they lay in a basis of the known Gauss’ method (method of exception of unknown
values) for solution of system of linear equations .
Elementary transformations of a matrix are:
1) rearrangement of two rows (columns);
2) multiplication of all row (column) elements of a matrix to some number, not equal to zero;
3) addition of two rows (columns) of the matrix multiplied by the same number, not equal to zero.
Two matrices are called equivalent if one of them is maybe received from another after final
number of elementary transformations. Generally equivalent matrices are not equal, but have the
same rank.
Calculation of determinants by means of elementary transformations
By means of elementary transformations it is easy to calculate a determinant of a matrix.
For example, it is required to calculate a determinant of the matrix:
where. a11  0
Then it is possible to bear a multiplier a11 :
now, subtracting from elements of the j-th column aij ( j  2) appropriating elements of the first
column, multiplied on a ij , we’ll receive the determinant:
which is equal to
where
and
Then we repeat the same actions for
and, if all elements
then
we’ll receive
If for any intermediate determinant
finally:
 n k
k
its left upper element a k 1 , k  1  0 it is necessary to
rearrange rows or columns in  nk so that a new left upper element will not be equal to zero. If Δ
≠ 0 it always can be made. Thus it is necessary to consider, that the sign on a determinant changes
depending on what element  pq is the main one (that is when the matrix is transformed so, that
a pq  1 ). Then the sign on an appropriating determinant is equal to ( 1) p  q .
E x a m p l e . By means of elementary transformations result the matrix
to a triangle type.
S o l u t i o n . First we’ll multiply the first row of the matrix by 4, and the second – by (–1) and
add the first row to the second:
Now we’ll multiply the first row by 6, and the third – by (–1) and add the first row to the third:
Finally, we’ll multiply the second row by 2, and the third – by (–9) and add the second row to the
third:
As a result the upper triangular matrix
is received.