Step 1:

Let the cost of one cycle =x

Let the cost of one bike =y

Let the cost of one car =z

According to the given information,the equation can be formed as follows:

3x+y+z=188000

2x+2y+z=228000

4x+2y+z=330000

This is of the form AX=B.

Where $A=\begin{bmatrix}3 & 1&1\\2 & 2 & 1\\4 & 2 &1\end{bmatrix},X=\begin{bmatrix}x\\y\\z\end{bmatrix}$ and $B=\begin{bmatrix}188000\\228000\\330000\end{bmatrix}$

Now first let us find the determinant of A,by expanding along $R_1$

$|A|=3(2\times 1-2\times 1)-1(2\times 1-4\times 1)+1(2\times 2-4\times 2)$

$\;\;=0+2-4=-2\neq 0.$

Hence $A^{-1}$ exists.

Step 2:

Next let us find the cofactors of A,

$A_{11}=(-1)_{1+1}\begin{vmatrix}2 & 1\\2 & 1\end{vmatrix}$=2-2=0.

$A_{12}=(-1)_{1+2}\begin{vmatrix}2 & 1\\4 & 1\end{vmatrix}$=-(2-4)=2

$A_{13}=(-1)_{1+3}\begin{vmatrix}2 & 2\\4 & 2\end{vmatrix}$=4-8=-4.

$A_{21}=(-1)_{2+1}\begin{vmatrix}1 & 1\\2 & 1\end{vmatrix}$=-(1-2)=1.

$A_{22}=(-1)_{2+2}\begin{vmatrix}3 & 1\\4 & 1\end{vmatrix}$=3-4=-1.

$A_{23}=(-1)_{2+3}\begin{vmatrix}3 & 1\\4 & 2\end{vmatrix}$=-(6-4)=-2.

$A_{31}=(-1)_{3+1}\begin{vmatrix}1 & 1\\2 & 1\end{vmatrix}$=1-2=-1

$A_{32}=(-1)_{3+2}\begin{vmatrix}3 & 1\\2 & 1\end{vmatrix}$=-(3-2)=-1.

$A_{33}=(-1)_{1+1}\begin{vmatrix}3 & 1\\2 & 2\end{vmatrix}$=6-2=4.

Hence the adjoint of A is $\begin{bmatrix}A_{11} & A_{21} & A_{31}\\A_{12} & A_{22} & A_{32}\\A_{13} & A_{23} & A_{33}\end{bmatrix}$

$\qquad\qquad\qquad\qquad=\begin{bmatrix}0 & 1 & -1\\2 & -1 & -1\\-4& -2& 4\end{bmatrix}$

$A^{-1}=\frac{1}{|A|}adj(A)$,we know |A|=-2.

$A^{-1}=\frac{1}{-2}\begin{bmatrix}0 & 1 & -1\\2 & -1 & -1\\-4 & -2 & 4\end{bmatrix}$

Step 3:

$A^{-1}B=X$,substituting for $A^{-1}$,B and X we get

$\begin{bmatrix}x\\y\\z\end{bmatrix}=\frac{1}{-2}\begin{bmatrix}0 & 1& -1\\2 &-1 & -1\\-4 & -2 & 4\end{bmatrix}\begin{bmatrix}188000\\228000\\330000\end{bmatrix}$

$\begin{bmatrix}x\\y\\z\end{bmatrix}=\frac{-1}{2}\begin{bmatrix}0+228000-330000\\376000-228000-330000\\-752000-456000+1320000\end{bmatrix}$

$\begin{bmatrix}x\\y\\z\end{bmatrix}=\frac{-1}{2}\begin{bmatrix}-102000\\-182000\\-112000\end{bmatrix}$

x=51000

y=91000

z=56000

Cost of 1 cycle is Rs. 51000

Cost of 1 bike is Rs. 91000

Cost of 1 car is Rs. 56000