JEE Main 2025 (Online) 29th January Evening ShiftQuestion 1
Let $A=[a_{ij}]$ be a matrix of order $3\times3$, with $a_{ij}=(\sqrt2)^{i+j}$. If the sum of all the elements in the third row of $A^{2}$ is $\alpha+\beta\sqrt2$, $\alpha,\beta\in\mathbb Z$, then $\alpha+\beta$ is equal to :
Matrices
Solution
$A=vv^T$ with $v=(\sqrt2,2,2\sqrt2)^T$, so $A^2=14A$. Third row of $A$ sums to $4+4\sqrt2+8=12+4\sqrt2$; times $14$: $168+56\sqrt2$. $\alpha+\beta=224$.