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A matrix A is defined as non-singular if its determinant is non-zero, i.e., $|A| \neq 0$. This condition ensures that the inverse matrix A-1 exists. Thus, options (C) and (D) are true statements.
For any square matrix A, the relationship between the determinant of the adjoint and the original matrix is given by: $$|adj(A)| = |A|^{n-1}$$ where n is the order of the matrix. Since $|A| \neq 0$, it follows that $|adj(A)| \neq 0$. Therefore, adj(A) is also a non-singular matrix. This makes option (A) false.
The property $(adj A)^{-1} = adj(A^{-1})$ is a standard identity in matrix algebra for non-singular matrices. Since A is non-singular, A-1 exists, and the identity holds true. Thus, option (B) is true.
Final Answer: (A)