GATE | GATE-CS-2016 (Set 1) | Question 15

Two eigenvalues of a 3 x 3 real matrix P are (2 + √ -1) and 3. The determinant of P is _____   

(A)

0

(B)

1

(C)

15

(D)

-1

Answer: (C)

Explanation:

The determinant of a real matrix can never be imaginary. So, if one eigen value is complex, the other eigen value has to be its conjugate.   So, the eigen values of the matrix will be 2+i, 2-i and 3.   Also, determinant is the product of all eigen values. So, the required answer is (2+i)*(2-i)*(3) = (4-i²)*(3) = (5)*(3) = 15.   Thus, C is the required answer. 

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