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: Using determinant formula of 3 x 3 matrix as
A = a11 a12 a13
a21 a22 a23
a31 a31 a33
then, determinant of A will be calculated by using following formula
A = a11 ( a22 x a33 – a23 x a31) – a12 ( a21 x a33 – a23 x a31) +a13 (a21 x a31 - a22 x a31)
Using this formula, to find determinant of given matrix is given by
can be calculated as
1 2 3
A = 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 – As per this law, sum of current flows from different loops is equal
to the sum of current that meets at certain point on a node. To determine, unknown variables in a
closed loop, Kirchoff's second law has been used as Loop Current Analysis of following algebric
algorithms -
2 x + 3 y – 4 z = 26 ...(i)
x – 5 y – 3 z = -87 ...(ii)
-7 x + 2 y + 6 z = 12 ...(iii)
On substituting and simplifying these equations, we get
x = 26 – 3 y + 4 z from equation first,
2
where, by putting this value in equation second equation and simplifying, we get
y = 200 – 2z
13
Now, substituting this value of y in x, we get x = 49 z – 262
26
So, using both value of x and y in equation third to eliminate two variables and convert into
third, we will get -
then,
-7 (49 z – 262) + 2 (200 - 2z) + 6 z = 12
26 13
- 343 z + 1834 + 800 – 8z + 156 z = 312
-195 z = -2322
z = 11.90
so, x = 49 (11.90) – 262 = 12.35 and
26
y = 200 – 2z = 200 – 2 x 11.90 = 13.55
2