Findα>0andβ>0sothatthematrix
4α1
A = 2β 5 4 is strictly diagonally dom-
β2α
inant.
To determine if a matrix is strictly diagonally dominant, we need to ensure that the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row:
|4α| > |1| + |2β|
|5| > |2βα| + |4|
In order to satisfy this inequality, we need to ensure that |4α| > 1 + 2β and 5 > 2βα + 4.
From the first inequality, we can see that α must be greater than 1/4 and β must be greater than (|4α| - 1)/2.
From the second inequality, we can see that α can be any positive value since β can be adjusted to satisfy the inequality.
Therefore, α > 0 and β > (|4α| - 1)/2.