A diet contains x units of food P, 2 units of food Q and 3 units of food R. One unit of P, Q and R gives 50, 80 and 40 calories respectively. The quantities are written as the row matrix \(\begin{bmatrix}x&2&3\end{bmatrix}\) and the calories per unit as the column matrix \(\begin{bmatrix}50\\80\\40\end{bmatrix}\). If the product of the row matrix and the column matrix is \(\begin{bmatrix}430\end{bmatrix}\), then the value of x is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The product of a \(1 \times 3\) row matrix and a \(3 \times 1\) column matrix is a \(1 \times 1\) matrix; its element is the sum of the products of corresponding elements:
\( \displaystyle \begin{bmatrix}x&2&3\end{bmatrix}\begin{bmatrix}50\\80\\40\end{bmatrix} = \begin{bmatrix}50x + 160 + 120\end{bmatrix} \)
Equating with \(\begin{bmatrix}430\end{bmatrix}\):
\(50x + 280 = 430\)
\(50x = 150\)
\(x = 3\)
So the diet contains 3 units of food P.