multiply row 1 by two and add the result to row 3, and (c) subtract row 1 from row
2. Performing these calculations produces
|A|=
∣∣
∣∣
∣∣
∣∣
∣
1 0 0 2 6
0 1 0 1 3
0 0 1 1 3
0 −1 0 2 1
0 0 1 2 6
∣∣
∣∣
∣∣
∣∣
∣
.
Now perform the operations: (a) add row 2 to row 4 and (b) subtract row 3 from
row 5. The determinant now has the form
|A|=
∣∣
∣∣
∣∣
∣∣
∣
1 0 0 2 6
0 1 0 1 3
0 0 1 1 3
0 0 0 3 4
0 0 0 1 3
∣∣
∣∣
∣∣
∣∣
∣
.
Observe that the row operations performed have produced zeros both above and
below the main diagonal. Next perform the operations of (a) subtracting twice row
5 from row 1, (b) subtracting row 5 from row 2, (c) subtracting row 5 from row 3,
and (d) subtracting row 5 from row 4. These operations produce
|A|=
∣∣
∣∣
∣∣
∣∣
∣
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 2 1
0 0 0 1 3
∣∣
∣∣
∣∣
∣∣
∣
.
By expanding |A|using cofactors of the first rows and associated subdeterminants,
there results
|A|= (1)(1)(1)
∣∣
∣∣2 1
1 3
∣∣
∣∣= 5.
A much more general procedure for calculating the determinant of a matrix A
is to use row operations and reduce |A|= det(A)to a triangular form having all zeros
below the main diagonal. For example, reduce Ato the form:
|A|= det(A) =
∣∣
∣∣
∣∣
∣∣
∣∣
a 11 a 12 a 13... a 1 n
0 a 22 a 23... a 2 n
0 0 a 33... a 3 n
..
.
..
.
..
.
... ..
.
0 0 0... a nn
∣∣
∣∣
∣∣
∣∣
∣∣
.
