Question

If a is a 5x5 matrix with trace 15 and if 2 and 3 are eigen values of a, each with algebric multiplicity 2,then the determinat of a is equal to

Answer 1

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Answer 2

Answer: Determinant of matrix A is 180

Step-by-step explanation:

Order of matrix A  =  5 × 5

∴ Number of eigen values of matrix A = 5

Given,

2 is eigen value with algebraic multiplicity 2

3 is eigen value with algebraic multiplicity 2

Let the fifth eigen value be λ

Eigen values of A ⇒ 2, 2, 3, 3, λ

Trace of matrix = sum of eigen values

⇒  15   =  2 + 2 + 3 + 3 + λ

⇒   15  =  10 + λ

⇒  15 - 10 = λ

⇒   5  =  λ

Eigen values of matrix A ⇒ 2, 2, 3, 3, 5

Determinant of matrix A = Product of eigen values of A

Determinant of matrix A = 2 × 2 × 3 × 3 × 5

Determinant of matrix A = 180

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