If \( A=\begin{bmatrix}1 & -1\\2 & -1\end{bmatrix},\ B=\begin{bmatrix}x & 1\\y & -1\end{bmatrix} \) and \( (A+B)^{2}=A^{2}+B^{2} \), then which of the following are correct?
(A) \( x=1 \)
(B) \( x=4 \)
(C) \( y=1 \)
(D) \( y=4 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
Matrix multiplication is not commutative, so
\( \displaystyle (A+B)^{2}=A^{2}+AB+BA+B^{2} \)
Hence \( (A+B)^{2}=A^{2}+B^{2} \) exactly when \( AB+BA=O \).
\( \displaystyle AB=\begin{bmatrix}1 & -1\\2 & -1\end{bmatrix}\begin{bmatrix}x & 1\\y & -1\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}x-y & 2\\2x-y & 3\end{bmatrix} \)
\( \displaystyle BA=\begin{bmatrix}x & 1\\y & -1\end{bmatrix}\begin{bmatrix}1 & -1\\2 & -1\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}x+2 & -x-1\\y-2 & -y+1\end{bmatrix} \)
\( \displaystyle AB+BA=\begin{bmatrix}2x-y+2 & 1-x\\2x-2 & 4-y\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix} \)
From the (1, 2) and (2, 2) entries:
\( \displaystyle 1-x=0\ \Rightarrow\ x=1 \)
\( \displaystyle 4-y=0\ \Rightarrow\ y=4 \)
Check the (1, 1) entry: 2 − 4 + 2 = 0, and the (2, 1) entry: 2 − 2 = 0.
So x = 1 and y = 4.
Checking the statements
(A) \( x=1 \) — Correct. The (1, 2) entry gives x = 1.
(B) \( x=4 \) — Incorrect. x is 1, not 4.
(C) \( y=1 \) — Incorrect. y is 4, not 1.
(D) \( y=4 \) — Correct. The (2, 2) entry gives y = 4.
Tip: \( (A+B)^{2}=A^{2}+2AB+B^{2} \) holds only when AB = BA; in general expand it as \( A^{2}+AB+BA+B^{2} \).