Elementary transformations of matrices
Then it is possible to bear a multiplier :
now, subtracting from elements of the j -th column appropriating elements of the first column, multiplied on , we’ll receive the determinant:
which is equal to where
Then we repeat the same actions for and, if all elements then we’ll receive finally:
If for any intermediate determinant its left upper element it is necessary to rearrange rows or columns in 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 is the main one (that is when the matrix is transformed so, that ). Then the sign on an appropriating determinant is equal to .
to a triangle type.
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.
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