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Assume determinant of a $3\times3$ matrix is a prime number and trace is $15.$Largest eigen value is _______(Assume only positve eigen values)
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Determinant is prime means product of these eigen values (3 because matrix is of dimension 3) is prime. When can product of 3 numbers be prime? Only when one of them is prime and other two are 1.

If you take two of them prime then the product becomes a non prime no. having factors as 1, those two prime numbers you multiplied and the number itself, thereby violating the  conditions of being a prime no.

Let E1,E2,E3. And E1=E2=1.

E1+E2+E3=15 (given)

1+1+E3=15

E3=13.

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