Analyse and Model Engineering System

Analyse and Model Engineering
System

TASK 2.0
2.1 To find the determinant of 3x3 matrix -
1 2 3
0 -4 1
0 3 -1
Solution: Determinant of 3 x 3 matrix can be calculated by using formula -
A = a b c
d e f
g h i
then, determinant of A can be determined by using -
A = a (ei – fh) – b (di – fg) + c (dh - eg)
Let determinant of given matrix is given by A
i.e.
A = 1 2 3
0 -4 1
0 3 -1
then, determinant of A = [ 1 ( -4 x -1 – 1 x3) - 2 ( 0 x -1 – 1 x 0 ) + 3 (0 x 3 – 1 x 0) ]
= [ 1 (4 – 3) – 2 ( 0 – 0) + 3 ( 0 – 0 )]
= 1
1

2.2 Three closed loops of a D.C. Circuit is given by -
2 x + 3 y – 4 z = 26
x – 5 y – 3 z = -87
-7 x + 2 y + 6 z = 12
To determine current flow, by Kirchoff's laws
Solutions: Kirchoff's Laws – this law states that at any node or junction in an electrical circuit,
algebraic sum of entire flow of current meet at zero in a network of conductors. Therefore, using
this law, current flow in given closed loops of DC, can be determined by Using Loop Current
Analysis to find unknown variables -
2 x + 3 y – 4 z = 26 ...(i)
x – 5 y – 3 z = -87 ...(ii)
-7 x + 2 y + 6 z = 12 ...(iii)
Using substitution method,
From equation (i),
x = 26 – 3 y + 4 z ...(iv)
2
Put this value of x into equation (ii),
26 – 3 y + 4 z - 5 y – 3 z = -87
2
26 – 3 y + 4 z – 10 y – 6 z = -174
- 13 y – 2 z = -200
13 y = 200 – 2z
y = 200 – 2z
13
Put this value of y in equation (iv) then,
x = 26 – 3 (200 – 2z ) + 4 z
13
2
x = 338 – 600 – 3 z + 52 z
26
2