Question

If the product of two eigen values of the matrix \(\begin{bmatrix} 6 & -2 & 2 \\\ -2 & 3 & -1 \\\ 2 & -1 & 3 \end{bmatrix}\) is 16, then third eigen value is
1. 2
2. -2
3. 36
4. 6

Answer 1

Correct Answer - Option 1 : 2

Explanation:

\(\left[ A \right] = \left[ {\begin{array}{*{20}{c}} 6&{ - 2}&2\\ { - 2}&3&{ - 1}\\ 2&{ - 1}&3 \end{array}} \right]\)

As we know:

Product of Eigen values = Determinant of matrix

Determinant of matrix = 6 × (9 – 1) – (- 2) (- 6 + 2) + 2 (2 - 6)

Determinant of matrix = 48 – 8 – 8

∴ Determinant of matrix = 32

Now,

Let the third Eigen value be x

Product of Eigen value = Determinant of [A]

16 × x = 32

∴ x = 2

∴ Third Eigen value = 2