Matrix Properties Analysis

We analyze the given statements regarding a square matrix A:

• Statement A: If A is invertible, then $\text{det(A)} \neq 0$. This is a fundamental property of invertible matrices. A matrix is invertible if and only if its determinant is non-zero. Thus, Statement A is true.
• Statement B: If $\text{det(A)} = 0$, A has no inverse. This is the direct consequence of the property mentioned in Statement A. A matrix with a determinant of zero is called a singular matrix and does not have an inverse. Thus, Statement B is true.
• Statement C: All eigenvalues of A are always positive. This is incorrect. Eigenvalues can be zero, negative, or complex numbers, depending on the matrix. For example, the identity matrix has eigenvalues equal to 1 (positive), but a matrix like $\begin{pmatrix} -1 & 0 \\ 0 & -2 \end{pmatrix}$ has negative eigenvalues. Thus, Statement C is false.
• Statement D: $\text{A}^T\text{A}$ is always invertible if A is invertible. If A is invertible, its transpose $\text{A}^T$ is also invertible. The product of two invertible matrices is always invertible. Therefore, $\text{A}^T\text{A}$ is invertible. Thus, Statement D is true.

Conclusion

Statements A, B, and D are true, while Statement C is false.

Therefore, the correct option includes A, B, and D.