Given: D = diag [d₁, d₂, d₃]

It is also given that d₁ ≠ 0, d₂ ≠ 0, d₃ ≠ 0

A diagonal matrix D = diag(d₁, d₂, …d_n) is invertible iff all diagonal entries are non – zero, i.e. d_i ≠ 0 for 1 ≤ i ≤ n

If D is invertible then D^-1 = diag(d₁^-1, …d_n^-1)

By the Inverting Diagonal Matrices Theorem, which states that

Here, it is given that d₁ ≠ 0, d₂ ≠ 0, d₃ ≠ 0

∴ D is invertible

⇒ D^-1 = diag [d₁^-1, d₂^-1, d₃^-1]

Hence Proved.