Let's analyze each statement one by one. \[\] 1. \(|\text{adj}(A)| \neq 0\): For an invertible matrix \(A\), the adjugate (or adjoint) of \(A\), denoted \(\text{adj}(A)\), is also invertible, and thus \(|\text{adj}(A)| \neq 0\). This is always true because \(\text{adj}(A)\) has the same invertibility as \(A\). \[\] 2. \(|A| \neq 0\): Since \(A\) is invertible, it means that \(\det(A) = |A| \neq 0\). This is always true for an invertible matrix. \[\] 3. \(|A A^{-1}| = 1\): Since \(A^{-1}\) is the inverse of \(A\), the determinant of the product of \(A\) and \(A^{-1}\) is: \[ |A A^{-1}| = |A| \cdot |A^{-1}| = |A| \cdot \frac{1}{|A|} = 1 \] This is always true. \[\] 4. \(|A \text{adj}(A)| \neq 1\): The determinant of the product of \(A\) and \(\text{adj}(A)\) is given by: \[ |A \cdot \text{adj}(A)| = |A| \cdot |\text{adj}(A)| = |A| \cdot |A|^{n-1} = |A|^n \] where \(n\) is the size of the square matrix \(A\). This is generally not equal to 1 unless \(A\) has a very specific form. In general, this statement is not always true.