Theorems 21.1.23 - 21.1.25 can be used to find determinants using row operations. As pointed out above,
the method of Laplace expansion will not be practical for any matrix of large size. Here is an example in
which all the row operations are used.
times the third row added to the second row. By Theorem 21.1.23
this didn’t change the value of the determinant. Then the last row was multiplied by
(− 3)
. By Theorem
21.1.19 the resulting matrix has a determinant which is
(− 3)
times the determinant of the un-multiplied
matrix. Therefore, we multiplied by −1∕3 to retain the correct value. Now replace the last row with 2
times the third added to it. This does not change the value of the determinant by Theorem 21.1.23. Finally
switch the third and second rows. This causes the determinant to be multiplied by
You could do more row operations or you could note that this can be easily expanded along
the first column followed by expanding the 3 × 3 matrix which results along its first column.
Thus
times the first row added to it. Next take −2 times the first row
and add to the third and finally take −3 times the first row and add to the last row. This
yields
By Theorem 21.1.23 this matrix has the same determinant as the original matrix. Remember you can work
with the columns also. Take −5 times the last column and add to the second column. This
yields
which by Theorem 21.1.23 has the same determinant as the original matrix. Lets expand it now along the
first column. This yields the following for the determinant of the original matrix.
We suggest you do not try to be fancy in using row operations. That is, stick mostly to the one which
replaces a row or column with a multiple of another row or column added to it. Also note there is no way
to check your answer other than working the problem more than one way. To be sure you have gotten it
right you must do this.