Question

If the product of eigenvalues of the matrix \(A =\left[ {\begin{array}{*{20}{c}} {1}&{2}&{-1}\\ {3}&{5}&{2}\\ {1}&{k}&{2} \end{array}} \right]\) is -8, then the value of k will be: 
1. 3
2. 2
3. -2
4. -3

Answer 1

Correct Answer - Option 1 : 3

Concept:

Sum of elements along principle diagonal = Trace = ∑ Eigen values

Product of eigen values = determinant = det (A)

Calculation:

Given:

\(A =\left[ {\begin{array}{*{20}{c}} {1}&{2}&{-1}\\ {3}&{5}&{2}\\ {1}&{k}&{2} \end{array}} \right]\)

We know that;

Det (A) = product of eigen values

1 × (10 - 2 × k) - 2 × (6 -2) - (3 × k - 5) = -8

10 - 2k - 8 - 3k + 5 = -8

7 - 5k = -8

15 = 5k

k = 3