The cost of 8kg apple and 3kg is Rs 70. The cost of 10kg apple and 6kg orange is 90. Find cost of each item if x is cost of apples per kg and y is cost of oranges per kg.

The given situation can be represented by a system of equations as:
8x+3y=70
10x+5y=90
Consider, A=\(\begin{bmatrix}8&3\\10&5\end{bmatrix}\), X=\(\begin{bmatrix}x\\y\end{bmatrix}\), B=\(\begin{bmatrix}70\\90\end{bmatrix}\)
It can be expressed in the form of AX=B
?X=A^-1 B
We know that, A^-1=\(\frac{1}{|A|}\) adj A=\(\frac{1}{10} \begin{bmatrix}5&-3\\-10&8\end{bmatrix}\)
?X=\(\frac{1}{10} \begin{bmatrix}5&-3\\-10&8\end{bmatrix}\begin{bmatrix}70\\90\end{bmatrix}\)=\(\frac{1}{10} \begin{bmatrix}5×70-3×90\\-10×70+8×90\end{bmatrix}\)=\(\frac{1}{10} \begin{bmatrix}300-270\\-700+720\end{bmatrix}\)=\(\frac{1}{10} \begin{bmatrix}30\\20\end{bmatrix}\)
?x=3, y=2.
i.e. The cost of apples is Rs 3 per kg and the cost of oranges is Rs 2 per kg.