Find the number of integer numbers of a such that the matrix ( − 1 0 5 0 a − 50 0 5 0 6 − a ) is negative definite.For that, the eigenvalues must be negative, right?So all these have to be negative and we get a system of inequalities: a − 50 &lt; 0 1 2 ( − a 2 − 14 a + 149 − a + 5 ) &lt; 0 1 2 ( a 2 − 14 a + 149 − a + 5 ) &lt; 0So we get from the first a &lt; 50, from the second a ∈ R and from the last one a &gt; 31.So the integer values for a are from 32 to 49, so there are 49 − 32 + 1 = 18 values, right?

Question

Find the number of integer numbers of   a such that the matrix      (                            −          1                          0                          5                                      0                          a          −          50                          0                                      5                          0                          6          −          a                      )  is negative definite.For that, the eigenvalues must be negative, right?So all these have to be negative and we get a system of inequalities:  a  −  50  &amp;lt;  0        1    2        (    −                  a        2            −      14      a      +      149        −    a    +    5    )    &amp;lt;  0        1    2        (                  a        2            −      14      a      +      149        −    a    +    5    )    &amp;lt;  0So we get from the first   a  &amp;lt;  50, from the second   a  ∈  R and from the last one   a  &amp;gt;  31.So the integer values for a are from   32 to   49, so there are   49  −  32  +  1  =  18 values, right?

Beckham Leach · Accepted Answer

Your approach is correct, but it could be slightly simpler: with the help of the rule of Sarrus or by other means, it&#039;s easy to find the characteristic polynomial is:  (  λ  −  a  +  50  )  (      λ    2    +  (  a  −  5  )  λ  +  a  −  31  )The eigenvalues must all be negative, so, for the first factor, that means   a  &amp;lt;  50. For the second factor, instead of computing the roots, note that the sum of the roots must be negative, while the product of the roots must be positive. With Vieta&#039;s formulas, that means   a  &amp;gt;  31 and   a  &amp;gt;  5. So the condition is indeed   31  &amp;lt;  a  &amp;lt;  50.

homosepian9wlgl · Accepted Answer

Great, but maybe there is another solution?

Find the number of integer numbers of a such that the matrix ( <mtable rowspac

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