Determinant

By Md.Bilal Khan

DETERMINANT:

In linear algebra, the determinant is scalar value. Here we are just going to study what determinant
Exactly is? How it is used to solve simultaneous equations.
A determinant is a factor or cause that makes something to happen or leads directly to a decision.
As a noun or adjective it refers to determining some vale or something for given input.

Determinant is generally denoted by |A|, det(A) or det A.

Determinant can be positive or a negative number.
In our own words we can define determinant as within two vertical lines taking input as row or column and perform cross multiplication to get output for 2X2 determinant.
see this:

is a Determinant. (a, b),(c, d) are rows   and  are columns.

Degree of this determinant is 2, because there are 2 elements in each column and 2 elements in each row. Solution of this determinant represents a number which is (ad-bc).
Cross multiplication of elements.



∴ |A| = ad - bc.
Ad - bc is the value of determinant:
Determinants are usually represented with capital letters. Like A,B,C, D…..etc.

For example: = (5X9)-(3X7) = 45 - 21 = 24

Determinant method (Cramer’s Rule)
Using determinants, simultaneous equations can be solved easily and in less space. This method is known as determinant method. This method was first given by a Swiss mathematician Gabriel Cramer, so it is also known as Cramer’s method.

To use Cramer’s method, the equations are written as a1x+b1y= c1 and a2x + b2y = c2. Number the equations.


a1x + b1 y = c1 . . . (I)
a2x + b2y = c2 . . . (II)
Here x and y are variables, a1, b1, c1 and a2, b2, c2 are real numbers,
a1b2 - a2b1≠ 0
Now let us solve these equations.


Now in simple words we will try to understand solving of simultaneous equations with help of determinant. To solve simultaneous equations we need to solve three determinant of degree 2 after that we can easily get value of variable X, Y. Here we will use three terms D, Dx, Dy as determinant.

In determinant D we will use only coefficients of variable terms no constant term is allowed here.

D = = a1b2 - a2b1

Now to get Dx just ignore or at place of coefficients of variable x put constants.

Dx = = c1b2 - c2b1

Last is Dy at place of coefficients of variable of y put constants.

Dy = = a1c2 -a2c1

But our aim of solving this all is to get solution of simultaneous equations i.e to get value of X and y variable terms.

So ,

X =

Y =

(X ,Y) will be solution of simultaneous equations.

STEPS:

From the given simultaneous equations compute the value for D, Dx, Dy
Find D , Dx and Dy.
Finally use formula to find X, Y which is given above and find X,Y.

For example:

1) X+2Y = -1 ; 2X-3Y = 12
For solving above simultaneous equation using Cramers rule we will find D,Dx,Dy

D = = (1) x (-3) - (2) x (2) = -3 -4 = -7

For Dx at place of coefficient of variable X put constant terms

Dx = = (-1) x (-3) - (2) x (12) =

= 3 -24 = -21

For Dy at place of coefficient of variable Y put constant terms

Dy = = (1) x (12) - (-1) x (2)

= 12 + 2 = 14
We know that

X = = = 3

Y = = = -2

(X,Y) = (3,-2).