Question

Let A be a nonsingular diagonalizable matrix of order 3 with eigenvalues λ₁, λ₂, λ₃. Then A^-1 is diagonalizable if:
1. λ₁ = 2, λ₂ = 1, λ₃ = 3
2. λ1 = 1, λ2 = 2, λ3 = 0
3. λ1 = 0, λ2 = 0, λ3 = 0
4. λ1 = 2, λ2 = 0, λ3 = 0

Answer 1

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

Concept:

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

• The determinant of any square matrix is the multiplication of the eigenvalues of the given matrix.

Calculation: 

It is given that A is a non-singular diagonalizable matrix with eigenvalues λ1, λ2, λ3.

⇒  A^-1 = λ₁ × λ₂ × λ3       ---(1)

Since A is non-singular,

⇒ A^-1 ≠ 0      ---(2)

From equation (1) & (2), we get,

∴  λ1, λ2, λ₃ can't be zero.

So, from the options, only λ₁ = 2, λ₂ = 1 and λ₃ = 3 satisfies the above condition. 

Hence, λ1 = 2, λ2 = 1, λ3 = 3