The numbers of units of two products X and Y sold by two shops P and Q in January and February are given by the matrices
\(A = \begin{bmatrix}30&46\\24&52\end{bmatrix}\) (January) and \(B = \begin{bmatrix}42&38\\36&40\end{bmatrix}\) (February),
where in each matrix the first row is for shop P, the second row is for shop Q, the first column is for product X and the second column is for product Y. The average monthly sale of product Y at shop Q over the two months, given by the matrix \(\displaystyle\frac{1}{2}(A + B)\), is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 46
Matrices are added element by element, and a scalar multiplies every element:
\( \displaystyle A + B = \begin{bmatrix}72&84\\60&92\end{bmatrix} \)
\( \displaystyle \frac{1}{2}(A + B) = \begin{bmatrix}36&42\\30&46\end{bmatrix} \)
Shop Q is the second row and product Y is the second column, so the average monthly sale is 46.
Why the other options are incorrect
Option (2) 42 — Incorrect. It is the average for product Y at shop P (first row).
Option (3) 30 — Incorrect. It is the average for product X at shop Q (first column).
Option (4) 92 — Incorrect. It is the total sale \(52 + 40\) for the two months, not halved.