Q.A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds. Pack-I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds. Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds. Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds. The cost of 50 gm of cashew nuts is 50, 50 gm of raisins is 10, and 50 gm of almonds is ` 60. What is the cost of each gift pack?

Three basic operations are defined on matrices:

1. Scalar multiplication. For $A=[a_{ij}]_{m\times n}$ and a scalar $k$, $kA = [k\,a_{ij}]_{m\times n}$ — every entry is scaled by $k$. Taking $k=-1$ gives $-A = [-a_{ij}]$, the negative of $A$ (never call it "a negative matrix").

2. Addition and subtraction. These are defined only when $A$ and $B$ have the same order. Then $A+B = [a_{ij}+b_{ij}]$ and $A-B = A+(-1)B = [a_{ij}-b_{ij}]$, entrywise. If the orders differ, $A+B$ and $A-B$ are simply not defined. Addition/subtraction extends to any finite number of matrices of the same order.

3. Multiplication. $A$ is conformable for the product $AB$ only when the number of columns of $A$ equals the number of rows of $B$: if $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ exists and has order $m\times p$ — schematically $(m\times n)(n\times p) = (m\times p)$, with the inner dimensions matching and cancelling. Each entry of $AB$ is a row-by-column dot product: …