Question

Let A be a non-singular diagonalisable matrix of order 3 with eignvalues λ1, λ2, λ3. A^-1 is diagonalisable if:
1. λ₁ = 2,λ₂ = 0, λ₃ = -1
2. λ₁ = 0, λ₂ = 3, λ₃ = -2
3. λ₁ = -1, λ₂ = 2,λ₃ = -3
4. λ₁ = -3, λ₂ = 1, λ₃ = 0

Answer 1

Correct Answer - Option 3 : λ₁ = -1, λ₂ = 2,λ₃ = -3

Concept:

The determinant of any square matrix is the multiplication of the eigenvalue of a given matrix.

A non-singular matrix is a square matrix whose determinant is not zero.

A singular matrix is a square matrix whose determinant is zero.

Observation:

A is a non-singular diagonalizable matrix

Eigenvalues of A matrix =  λ₁, λ₂, λ₃.

Deteminant of matrix A = λ₁ × λ₂ × λ₃

As given the A, the matrix is non-singular

\(\left[ A \right] \ne 0\)

[A] = λ1 × λ2 × λ3 

So, λ1, λ2, λ₃ cannot be zero.

λ1 = -1, λ2 = 2,λ3 = -3 satisfy the solution.

Hence, option 3 is correct.