Theorems 19.2.23 - 19.2.25 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 19.2.23 this didn’t change the value of the determinant. Then the last
row was multiplied by
(− 3)
. By Theorem 19.2.19 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 19.2.23. 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
||11 22 ||
det(D ) = 1 (− 3)||14 − 17 || = 1485
and so det
(C )
= −1485 and det
(A)
= det
(B )
=
(−1)
3
(− 1485)
= 495.
Example 19.2.29Find the determinant of the matrix
( )
1 2 3 2
|| 1 − 3 2 1 ||
( 2 1 2 5 )
3 − 4 1 2
Replace the second row by
(− 1)
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 19.2.23 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 19.2.23 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.
Unfortunately, this process can go on and on when you keep getting different answers.
This is a good example of something you are better off using a computer algebra system
to find.