Construct a 2 2 matrix A = [a_ ij ] whose element a_ ij is given by: (i+j)^ 2 2

Let A = [a_ ij ]_ 2 2 So, the elements in a 2 2 matrix are a_ 11 , a_ 12 , a_ 21 , a_ 22 , A = a_ 11 &a_ 12 _ 21 &a_ 22 ...(i) a_ 11 = (1+1 )^22 = 42 = 2 a_ 12 = (1+2)^ 2 2 = 3^ 2 2 =9 2 a_ 21 = (2+1)^ 2 2 = 3^ 2 2 =9 2 a_ 22 = (2+2)^ 2 2 = 4^ 2 2 =16 2 = 8 So, (i) becomes A = ( cc 2 & 92 92 & 8 )

Construct a 2 2 matrix A = [a_ ij ] whose element a_ ij is given …

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Question

Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose element $a_{ij} $ is given by: $\frac{(i+j)^{2}}{2}$

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Answer

Let $A = [a_{ij}]_{2\times2}$
So, the elements in a $2\times2$ matrix are
$a_{11}, a_{12}, a_{21}, a_{22},$
$A = \begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} ...(i)$
$a_{11} =\frac{\left(1+1\right)^2}2 = \frac42 = 2$
$a_{12} = \frac{(1+2)^{2}}{2}=\frac{3^{2}}{2}=\frac{9}{2} $
$a_{21} = \frac{(2+1)^{2}}{2}=\frac{3^{2}}{2}=\frac{9}{2}$
$a_{22} = \frac{(2+2)^{2}}{2}=\frac{4^{2}}{2}=\frac{16}{2} = 8$
So, $(i)$ becomes
$A = \left(\begin{array}{cc} {2} & {\frac 92} \\ {\frac92} & {8} \end{array}\right)$

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