Question

verify the following system of linear equations is diagonally dominant or not x+2y+z=4,3x+4y+8z=13,x+3y+z=5​

Answer 1

Given : x+2y+z=4,3x+4y+8z=13,x+3y+z=5

To Find : system of linear equations is diagonally dominant or not

Solution:

x    + 2y   +  z  = 4

3x  + 4y   + 8z  = 13

x  + 3y   + z    =    5  

$\left[\begin{array}{ccc}1&2&1\\3&4&8\\1&3&1\end{array}\right]$

diagonally dominant :

If in every row of the matrix, the magnitude of the diagonal entry  is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row.  

aii ≥ ∑aij   i≠j      

a11 ≥  a12 + a13    

a11   = 1  , a12 + a13     = 2 + 1 = 3

Hence a11  < a12 + a13   so Matrix is not Diagonal dominant

a22  ≥  a21 + a23

a22 = 4    ,  a21 + a23 =  3 + 8 = 11

a22 < a21 + a23

a33 ≥  a31 + a32

a33 =  1 , a31 + a32 = 1 + 3 = 4

a33 <  a31 + a32

Hence system of linear equations is not diagonally dominant

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diagonally dominant or not

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