If $A=\begin{bmatrix}1 & 3 & x^2\\0& 1& 0\\0 & 0 & 1\end{bmatrix}, B=\begin{bmatrix}1 & -3 & y^2\\0& 1& 0\\0 & 0 & 1\end{bmatrix}$ and $AB=I_y,$ where $I_3$ is the identity matrix of order $3×3$, then $x^2+y^2$ equals :

0

The correct answer is Option (2) → 0

$AB=\begin{bmatrix}1 & 3 & x^2\\0& 1& 0\\0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & -3 & y^2\\0& 1& 0\\0 & 0 & 1\end{bmatrix}=\begin{bmatrix}1 & 0 & 0\\0& 1& 0\\0 & 0 & 1\end{bmatrix}$

$⇒AB=\begin{bmatrix}1 & 0 & x^2+y^2\\0& 1& 0\\0 & 0 & 1\end{bmatrix}=\begin{bmatrix}1 & 0 & 0\\0& 1& 0\\0 & 0 & 1\end{bmatrix}$

$⇒x^2+y^2=0$