The values of \( x \) and \( y \) if X = Y, where
\( X=\begin{bmatrix}x-2 & 3y\\6 & -6\end{bmatrix},Y=\begin{bmatrix}2x+1 & y^2+2\\6 & y^2-5y\end{bmatrix} \) are:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( x=-3 \) and \( y=2 \)
Two matrices are equal only when every pair of corresponding entries is equal.
From the (1, 1) entries:
\( \displaystyle x-2=2x+1 \)
\( \displaystyle x=-3 \)
From the (1, 2) entries:
\( \displaystyle 3y=y^2+2 \)
\( \displaystyle y^2-3y+2=0 \)
\( \displaystyle (y-1)(y-2)=0 \)
\( \displaystyle y=1\ \text{or}\ y=2 \)
From the (2, 2) entries:
\( \displaystyle -6=y^2-5y \)
\( \displaystyle y^2-5y+6=0 \)
\( \displaystyle (y-2)(y-3)=0 \)
\( \displaystyle y=2\ \text{or}\ y=3 \)
y must satisfy both equations, so y = 2. Hence x = −3 and y = 2.
Why the other options are incorrect
Option (1) \( x=-3 \) and \( y=3 \) — Incorrect. y = 3 satisfies the (2, 2) entries but not the (1, 2) entries: 3y = 9, while \( y^2+2=11 \).
Option (3) \( x=2 \) and \( y=3 \) — Incorrect. x = 2 does not satisfy x − 2 = 2x + 1 (0 ≠ 5).
Option (4) \( x=-3 \) and \( y=1 \) — Incorrect. y = 1 satisfies the (1, 2) entries but not the (2, 2) entries: \( y^2-5y=-4 \), not −6.