Let A = [ a i j ] \mathrm{A}=\left[a_{i j}\right] A = [ a ij ] be a matrix of order 3 × 3 3 \times 3 3 × 3 , with a i j = ( 2 ) i + j a_{i j}=(\sqrt{2})^{i+j} a ij = ( 2 ) i + j . If the sum of all the elements in the third row of A 2 A^2 A 2 is α + β 2 , α , β ∈ Z \alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z} α + β 2 , α , β ∈ Z , then α + β \alpha+\beta α + β is equal to :
Step-by-Step Verified Solution
a m p ; A = [ ( 2 ) 2 a m p ; ( 2 ) 3 a m p ; ( 2 ) 4 ( 2 ) 3 a m p ; ( 2 ) 4 a m p ; ( 2 ) 5 ( 2 ) 4 a m p ; ( 2 ) 5 a m p ; ( 2 ) 6 ] a m p ; A = [ 2 a m p ; 2 2 a m p ; 4 2 2 a m p ; 4 a m p ; 4 2 4 a m p ; 4 2 a m p ; 8 ] a m p ; A 2 = 2 2 [ 1 a m p ; 2 a m p ; 2 2 a m p ; 2 a m p ; 2 2 2 a m p ; 2 2 a m p ; 4 ] [ 1 a m p ; 2 a m p ; 2 2 a m p ; 2 a m p ; 2 2 2 a m p ; 2 2 a m p ; 4 ] \begin{aligned}& A=\left[\begin{array}{lll}(\sqrt{2})^2 & (\sqrt{2})^3 & (\sqrt{2})^4 \\(\sqrt{2})^3 & (\sqrt{2})^4 & (\sqrt{2})^5 \\(\sqrt{2})^4 & (\sqrt{2})^5 & (\sqrt{2})^6\end{array}\right] \\& A=\left[\begin{array}{ccc}2 & 2 \sqrt{2} & 4 \\2 \sqrt{2} & 4 & 4 \sqrt{2} \\4 & 4 \sqrt{2} & 8\end{array}\right] \\& A^2=2^2\left[\begin{array}{ccc}1 & \sqrt{2} & 2 \\\sqrt{2} & 2 & 2 \sqrt{2} \\2 & 2 \sqrt{2} & 4\end{array}\right]\left[\begin{array}{ccc}1 & \sqrt{2} & 2 \\\sqrt{2} & 2 & 2 \sqrt{2} \\2 & 2 \sqrt{2} & 4\end{array}\right]\end{aligned} am p ; A = ( 2 ) 2 ( 2 ) 3 ( 2 ) 4 am p ; ( 2 ) 3 am p ; ( 2 ) 4 am p ; ( 2 ) 5 am p ; ( 2 ) 4 am p ; ( 2 ) 5 am p ; ( 2 ) 6 am p ; A = 2 2 2 4 am p ; 2 2 am p ; 4 am p ; 4 2 am p ; 4 am p ; 4 2 am p ; 8 am p ; A 2 = 2 2 1 2 2 am p ; 2 am p ; 2 am p ; 2 2 am p ; 2 am p ; 2 2 am p ; 4 1 2 2 am p ; 2 am p ; 2 am p ; 2 2 am p ; 2 am p ; 2 2 am p ; 4
a m p ; = 4 [ − a m p ; − a m p ; − − a m p ; − a m p ; − ( 2 + 4 + 8 ) a m p ; ( 2 2 + 4 2 + 8 2 ) a m p ; ( 4 + 8 + 16 ) ] a m p ; Sum of elements of 3 rd row = 4 ( 14 + 14 2 + 28 ) a m p ; a m p ; = 4 ( 42 + 14 2 ) a m p ; = 168 + 56 2 a m p ; α + β 2 ∴ a m p ; α + β = 168 + 56 = 224 \begin{aligned}&=4\left[\begin{array}{ccc}- & - & - \\- & - & - \\(2+4+8) & (2 \sqrt{2}+4 \sqrt{2}+8 \sqrt{2}) & (4+8+16)\end{array}\right]\\&\text { Sum of elements of } 3^{\text {rd }} \text { row }=4(14+14 \sqrt{2}+28)\\&\begin{aligned}& =4(42+14 \sqrt{2}) \\& =168+56 \sqrt{2} \\& \alpha+\beta \sqrt{2} \\\therefore \quad & \alpha+\beta=168+56=224\end{aligned}\end{aligned} am p ; = 4 − − ( 2 + 4 + 8 ) am p ; − am p ; − am p ; ( 2 2 + 4 2 + 8 2 ) am p ; − am p ; − am p ; ( 4 + 8 + 16 ) am p ; Sum of elements of 3 rd row = 4 ( 14 + 14 2 + 28 ) am p ; ∴ am p ; = 4 ( 42 + 14 2 ) am p ; = 168 + 56 2 am p ; α + β 2 am p ; α + β = 168 + 56 = 224