Fsc Part 1 Mathematics (Complete Solution)
Q7
Evaluate the following determinants:
(i) |
|
3 |
4 |
2 |
7 |
2 |
5 |
0 |
3 |
1 |
2 |
-3 |
5 |
4 |
1 |
-2 |
6 |
|
|
|
|
SOLUTION: |
|
3 |
4 |
2 |
7 |
2 |
5 |
0 |
3 |
1 |
2 |
-3 |
5 |
4 |
1 |
-2 |
6 |
|
|
|
|
= |
|
0 |
-2 |
11 |
-8 |
0 |
1 |
6 |
-7 |
1 |
2 |
-3 |
5 |
0 |
-7 |
10 |
14 |
|
|
|
R1-3R3 R2-2R3 R4-4R3 |
Expand by C1
=1(-1)1+3 |
|
-2 |
11 |
-8 |
1 |
6 |
-7 |
-7 |
10 |
-14 |
|
|
|
R1-3R3 R2-2R3 R4-4R3 |
=-[(-23)(63)+(22)(52)]
= -(-1449+1144)
= 305
(ii) |
|
2 |
3 |
1 |
-1 |
4 |
0 |
2 |
1 |
5 |
2 |
-1 |
6 |
3 |
-7 |
2 |
-2 |
|
|
|
|
SOLUTION: |
|
2 |
3 |
1 |
-1 |
4 |
0 |
2 |
1 |
5 |
2 |
-1 |
6 |
3 |
-7 |
2 |
-2 |
|
|
|
|
= |
|
0 |
0 |
1 |
0 |
0 |
-6 |
2 |
3 |
7 |
5 |
-1 |
5 |
-1 |
-13 |
2 |
0 |
|
|
|
C1-2C3 C2-3C3 C4-C3 |
Expand by R1
=1(-1)1+3 |
|
R1-3R3 R2-2R3 R4-4R3 |
=3(-91+15)
= 3(-76)
= 288
(iii) |
|
-3 |
9 |
1 |
1 |
0 |
3 |
-1 |
2 |
9 |
7 |
-1 |
1 |
-2 |
0 |
1 |
-1 |
|
|
|
|
SOLUTION: |
|
-3 |
9 |
1 |
1 |
0 |
3 |
-1 |
2 |
9 |
7 |
-1 |
1 |
-2 |
0 |
1 |
-1 |
|
|
|
|
= |
|
0 |
0 |
0 |
1 |
6 |
-15 |
-3 |
2 |
12 |
-2 |
-2 |
1 |
-5 |
9 |
2 |
-1 |
|
|
|
C1-3C4 C2-3C4 C3-C4 |
Expand by R1
=(-1)(3)(2) |
|
Taking Common 3 from R1 and 2 from R2 |
=6(-4+4)
= 6(0)
= 0