Cramer's rule

System of non homogeneous Linear equations(Cramer's rule)

 

ax+by+cz=0----homogeneous

ax+by+cz+d=0--------------------non homogeneous

 

Let 'n ' linear simultaneous equation in 'n' unknown x1,x2,x3............xn be

a11x1+a12x2+......................................a1nxn=b1

a21x1+a22x2+......................................a2nxn=b2

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an1x1+an2x2+......................................annxn=bn

 

let Δ= \begin{vmatrix} a11&a12 &......a1n & \\ a21 &a22 & ......a2n& \\ .& .& &. \\ an1&an2 &ann & \end{vmatrix}    ≠0

 

suppose A11,A12 ,A13...................etc denotes the cofactors of a11, a12,a13.............. multiply given equation respectively by A11, A21, ................An1and we get 

 

A11(a11x1+a12x2+......................................a1nxn=b1)

A21(a21x1+a22x2+......................................a2nxn=b2)

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An1(an1x1+an2x2+......................................annxn=bn)

 

that is,

         x1(a11A11+ a21A21+......................an1An1​) + x2(0)+x3(0)                            =b1A11+b2A21+..............bnAn1

⇒x1Δ=Δ1

x1=Δ1/Δ

where  Δ1 isobtained by replacing elements in the first column of  Δ bythe elements b1,b2...................bn

Similarly,

we get x2=Δ2/Δ

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.xn=Δ2/Δ

 

Example:

solve by Cramer's rule

x+2y+3z=6

2x+4y+z=7

3x+2y+9z=14

 

 

solution

we have to first find the values of Δ i.e cofficent of x ,y,z

Δ=\begin{vmatrix} 6&2 &3& \\ 7 &4 & 1& \\ 14& 2& 9&\\ & \end{vmatrix}\begin{vmatrix} 1&2 &3& \\ 2 &4 & 1& \\ 3& 2& 9&\\ & \end{vmatrix}

 

Δ=-20

 

Δ1\begin{vmatrix} 6&2 &3& \\ 7 &4 & 1& \\ 14& 2& 9&\\ & \end{vmatrix}

=-20

x1

= -20/-20

=1

 

 

Δ2=\begin{vmatrix} 1&6 &3& \\ 2 &7 & 1& \\ 3& 14& 9&\\ & \end{vmatrix}

=-20

y=Δ2

=-20/-20

=1

similarly

Δ3=\begin{vmatrix} 1&2 &6& \\ 2 &4 & 7& \\ 3& 2& 14&\\ & \end{vmatrix}

=-20

z=Δ3/Δ

=1

 

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