If \(X + Y = \begin{bmatrix}5&2\\0&9\end{bmatrix}\) and \(X - Y = \begin{bmatrix}3&6\\0&-1\end{bmatrix}\), then Y is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\begin{bmatrix}1&-2\\0&5\end{bmatrix}\)
Subtracting the second equation from the first removes X:
\((X + Y) - (X - Y) = 2Y\)
\( \displaystyle 2Y = \begin{bmatrix}5 - 3&2 - 6\\0 - 0&9 + 1\end{bmatrix} = \begin{bmatrix}2&-4\\0&10\end{bmatrix} \)
\( \displaystyle Y = \begin{bmatrix}1&-2\\0&5\end{bmatrix} \)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}4&4\\0&4\end{bmatrix}\) — Incorrect. This is X, found by adding the two equations and halving.
Option (2) \(\begin{bmatrix}-1&2\\0&-5\end{bmatrix}\) — Incorrect. This is \(-Y\); it comes from subtracting in the wrong order.
Option (3) \(\begin{bmatrix}2&-4\\0&10\end{bmatrix}\) — Incorrect. This is 2Y; it still has to be divided by 2.