Question

Let a be a 5x5 matrix with real entries such that the sum of the entries in each row of a is 1. then the sum of all the entries in a^3 is

Answer 1

Answer:

5x5=25 1/25a^3 +65 answer

Answer 2

Answer:

Sum of all entries in a^3 = 5

Step-by-step explanation:

The eigen vectors(let e) of matrix A is =  $\left[\begin{array}{ccc}1\\1\\1\\1\\1\end{array}\right]$ or $\left[\begin{array}{ccc}-1\\-1\\-1\\-1\\-1\end{array}\right]$

and corresponding to eigen value = 1

ae = e

The sum of all entries in a^3 is

$e^Ta^3e = e^Ta^2(ae)$

         $= e^Ta^2e$

         $= e^TAe = e^Te$

         $= n$

here n = 5

The sum of all entries in a^3 = n = 5

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