Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 2 DETERMINANTS
Proof: Expanding the determinant by the row whose terms are binomials, we
will see property (6) immediately.
(7) If we add to the elements of a row (or column) any constant multiple of the
corresponding elements in any other row (or column), the value of the determinant
is unaltered.
Proof: Applying property (6) to the determinant that results from the given
addition, we obtain a sum of two determinants: one is the original determinant
and the other contains two proportional rows. Then by property (4), the second
determinant is zero, and the proof is complete.
It is advisable to simplify a determinant before evaluating it. This may be done
with the help of properties (7) and (2), as shown in the following example.
Example A2.6
Evaluate
1 24 21 93
2 37 1 194
D ˆ
:
2 35 0 171
3 177 63 234
To simplify this, we want the ®rst elements of the second, third and last rows all to
be zero. To achieve this, add the second row to the third, and add three times the
®rst to the last, subtract twice the ®rst row from the second; then develop the
resulting determinant by the ®rst column:
1 24 21 93
0 85 43 8
85 43 8
D ˆ
ˆ
2 1 23
:
0 2 1 23
0 249 126 513
249 126 513
We can simplify the resulting determinant further. Add three times the ®rst row to
the last row:
85 43 8
D ˆ
2 1 23
:
6 3 537
Subtract twice the second column from the ®rst, and then develop the resulting
determinant by the ®rst column:
1 43 8
1 23
D ˆ
0 1 23
ˆ 3 537 ˆ537 23 …3† ˆ468:
0 3 537
546