Nomography Main > Determinant 4
ρ=Σ+Ψ
Basic Nomographic Form
The difference between a mere determinant and a nomogram determinant is that
the determinant is in Basic Nomographic Form. When it is in BNF the
determinant can be used to draw the scales of a nomogram, and the points
representing solution values are guaranteed to be on the same straight line.
A determinant is in Basic Nomographic Form if all of the following are true:
 The absolute value of the determinant is zero.
 Each variable is confined to one row only
(though in Grid Nomograms two variables can share a row).
 The last column must contain only +1.
You don't have to worry about #1, it should have been taken care of by the
insertion of the equation into the determinant. #2 is the reason one can use each
row to get the X coord and Y coord equations. Obviously if a variable is
on two or three rows the ownership of the coord equations is unclear.
So, to confine each variable to a row and to make the last column
composed of +1, one uses The Four Rules of Determinants.
The Four Rules of Determinants
RULE 1
Rows may be changed into columns and columns into rows.
i.e., one can rotate the determinant around the left diagonal axis. Example:
A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
A_{1} 
A_{2} 
A_{3} 
B_{1} 
B_{2} 
B_{3} 
C_{1} 
C_{2} 
C_{3} 

RULE 2
Each element of any row or any column may be multiplied by the same
quantity or divided by the same quantity.
Actually, this multiplies the value of the determinant by the quantity.
However, since the initial value of the determinant is zero, multiplying it
by any quantity will leave it unchanged. Examples:
A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
A_{1}M 
B_{1} 
C_{1} 
A_{2}M 
B_{2} 
C_{2} 
A_{3}M 
B_{3} 
C_{3} 

A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3}/M 
B_{3}/M 
C_{3}/M 

RULE 3
Any two rows or any two columns may be interchanged.
Actually this changes the sign of the value of the determinant. However,
since the initial value is zero, the sign is unchanged. Examples:
A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
B_{1} 
A_{1} 
C_{1} 
B_{2} 
A_{2} 
C_{2} 
B_{3} 
A_{3} 
C_{3} 

A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
A_{1} 
B_{1} 
C_{1} 
A_{3} 
B_{3} 
C_{3} 
A_{2} 
B_{2} 
C_{2} 

RULE 4
To the elements of any row or column may be added the corresponding elements
of any one or more rows or columns. To the elements of any row or column may be added the
same multiples of the corresponding elements
of any one or more rows or columns.
Examples:
A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
B_{1}+A_{1} 
B_{1} 
C_{1} 
B_{2}+A_{2} 
B_{2} 
C_{2} 
B_{3}+A_{3} 
B_{3} 
C_{3} 

A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 

= 
A_{1} 
B_{1} 
C_{1} 
A_{2} 
B_{2} 
C_{2} 
3A_{1}+A_{3} 
3B_{1}+B_{3} 
3C_{1}+C_{3} 

Alas, much like factoring, there is no set way to use the Four Rules to
convert a determinant into Basic Nomographic Form. One just has to play with it until it
works. Of course with experience, one will gain the skill to intuitively
see the steps required.
Example 1. Take the following determinant and convert it into Basic
Nomographic Form by using the Four Rules.
Using Rule 4, add column 2 to column 1:
1 
0 
v 
1 
1 
w 
u1 
1 
u^{2} 
Using Rule 2, divide the last row by u1
1 
0 
v 
1 
1 
w 
1 
1/(u1) 
u^{2}/(u1) 
Using Rule 3, interchange column 1 and column 3
0 
v 
1 
1 
w 
1 
1/(u1) 
u^{2}/(u1) 
1 
That is in Basic Nomographic Form, but just to be neat one can
use Rule 3 and interchange row 1 and row 3:
1/(u1) 
u^{2}/(u1) 
1 
0 
v 
1 
1 
w 
1 
Example 2. Take the following determinant and convert it into Basic
Nomographic Form by using the Four Rules.
logV 
0 
1 
logH 
1 
0 
log(π/4)+2logD 
1 
1 
Using Rule 2 multiply right column by 2:
logV 
0 
2 
logH 
1 
0 
log(π/4)+2logD 
1 
2 
Using Rule 4 add middle column to right column:
logV 
0 
2 
logH 
1 
1 
log(π/4)+2logD 
1 
1 
Using Rule 2 divide top row by 2:
logV/2 
0 
1 
logH 
1 
1 
log(π/4)+2logD 
1 
1 
That is in Basic Nomographic Form. However, the parallel scales will be
lying on their side instead of upright. Turn them upright by using
Rule 3 and interchanging the first and middle columns
0 
logV/2 
1 
1 
logH 
1 
1 
log(π/4)+2logD 
1 
Example 3. Take the following determinant and convert it into Basic
Nomographic Form by using the Four Rules.
(Note: " √ " should be a square root symbol)
√E 
V/8.03 
0 
0 
1 
1 
√W 
0 
1 
Using Rule 2 divide the middle column by V/8.03. This separates the variables into
individual rows:
√E 
1 
0 
0 
8.03/V 
1 
√W 
0 
1 
Using Rule 4 add the middle column to the right column to adjust the upper right element:
√E 
1 
1 
0 
8.03/V 
1+(8.03/V) 
√W 
0 
1 
Using Rule 2 divide the middle row by 1+(8.03/v):
√E 
1 
1 
0 
(8.03/V) / (1+(8.03/V)) 
1 
√W 
0 
1 
That is in Basic Nomographic Form.