Detailed Step-by-Step Vector Calculation:

• Method 1 (Direct Proportionality):
  $$\vec{B} = 6\hat{i} - 3\hat{j} + 6\hat{k} = 3(2\hat{i} - \hat{j} + 2\hat{k}) = 3\vec{A}$$
  Since $\vec{B}$ is a positive scalar multiple of $\vec{A}$ with scalar $k = 3 \gt 0$, the two vectors are collinear and point in the identical direction. Hence, the angle $\theta$ between them is $0^\circ$.
• Method 2 (Dot Product Formula):
  $$\cos\theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$$
  $$\vec{A} \cdot \vec{B} = (2)(6) + (-1)(-3) + (2)(6) = 12 + 3 + 12 = 27$$
  $$|\vec{A}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$$
  $$|\vec{B}| = \sqrt{6^2 + (-3)^2 + 6^2} = \sqrt{36 + 9 + 36} = \sqrt{81} = 9$$
  $$\cos\theta = \frac{27}{3 \times 9} = \frac{27}{27} = 1 \implies \theta = \cos^{-1}(1) = 0^\circ$$

Statement (A) is false because two non-zero matrices can have a product equal to the zero matrix ($AB = 0$ does not require $A = 0$ or $B = 0$). Statement (B) is true by definition: $|A| \neq 0$ means non-singular. Statement (C) is true: $\text{adj}(A^T) = (\text{adj}(A))^T$, so if $A^T = A$, then $\text{adj}(A)$ is symmetric. Statement (D) is true: the adjoint of a diagonal matrix is also diagonal. Option analysis: (A) Contains false statement (A). (B) (B), (C) and (D) only is the correct answer. (C) Contains false statement (A). (D) Contains false statement (A).