(1) Adjoint of symmetric matrix is symmetric
Let A be a symmetric matrix
then A^T = A.
we know that, (adj⁡A)^T = adj⁡A^T

⇒ (adj⁡A)^T = (adj⁡A)

∴ adj⁡A is symmetric matrix.
∴ Statement (2) is true.
(2) Adjoint of unit matrix is unit matrix.
Let A be a unit matrix
then A^T = A.
As we know that, adj⁡A^T = (adj⁡A)^T

 (adj A) = (adj⁡A)^T

∴ adjoint of unit matrix is unit. matrix.
∴ Statement 2 is true.
(3) A(aqjA) = (adj⁡A)A = |A|I

$\because {\mathrm{A}}^{-1}=\frac{\mathrm{adj}\mathrm{A}}{\left|\mathrm{A}\right|}$

multiplying both sides with A

$\Rightarrow {\mathrm{AA}}^{-1}=\frac{\mathrm{A}\left(\mathrm{adjA}\right)}{\left|\mathrm{A}\right|}$

∴ A(adjA) = |A|I

Similarly (adj A)A = |A|I
∴ statement 3 is true.
(4) Adjoint of a diagonal matrix is diagonal matrix. By properties of diagonal matrix it is clear that; adjoint of diagonal matrix is diagonal matrix
∴ statement: 4 is true.
⇒ all the four statements are true.
∴ None of these option D. is correct Answer.