(Solved): (a) The determinant of the sum of two matrices equals the sum of the determinants of the matrices. ...

(a) The determinant of the sum of two matrices equals the sum of the determinants of the matrices. False, consider the following matrices: \( \left[\begin{array}{ll}8 & 1 \\ 9 & 5\end{array}\right] \) and \( \left[\begin{array}{ll}8 & 1 \\ 9 & 5\end{array}\right] \). True, the determinant of the sum of an even number of matrices is equal to the sum of the determinants of those matrices. True, the determinant of the sum of any number of matrices equals the sum of the determinants of those matrices. False, consider the following matrices: \( \left[\begin{array}{ll}8 & 0 \\ 5 & 0\end{array}\right] \) and \( \left[\begin{array}{ll}5 & 0 \\ 8 & 0\end{array}\right] \). False, consider the following matrices: \( \left[\begin{array}{ll}8 & 8 \\ 8 & 2\end{array}\right] \) and \( \left[\begin{array}{ll}-8 & 8 \\ -8 & 2\end{array}\right] \). (b) If \( A \) and \( B \) are square matrices of order \( n_{s} \) and \( \operatorname{det}(A)=\operatorname{det}(B) \), then \( \operatorname{det}(A B)=\operatorname{det}\left(A^{2}\right) \). False, using the given the conditions \( \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)=\operatorname{det}(A) \operatorname{det}(A)=\operatorname{det}(2 A) \). True, using the given the conditions \( \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)=\operatorname{det}(A) \operatorname{det}(-A)=\operatorname{det}(A(-A))=\operatorname{det}\left(A^{2}\right) \). True, using the given the conditions \( \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)=\operatorname{det}(A) \operatorname{det}(A)=\operatorname{det}(A A)=\operatorname{det}\left(A^{2}\right) \). False, using the given the conditions \( \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)=\operatorname{det}(A)+\operatorname{det}(A)=2 \operatorname{det}(A)=\operatorname{det}(2 A) \). True, using the given the conditions \( \operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)=\operatorname{det}(A)+\operatorname{det}(A)=2 \operatorname{det}(A)=\operatorname{det}\left(A^{2}\right) \). (c) If the determinant of an \( n \times n \) matrix \( A \) is nonzero, then \( A \mathbf{x}=O \) has only the trivial solution. False, if the determinant of an \( n \times n \) matrix \( A \) is nonzero then the matrix is not invertible, which means that \( A \mathbf{x}=0 \) has infinite solutions. False, if the determinant of an \( n \times n \) matrix \( A \) is nonzero then the matrix is not invertible, which means that \( A x=0 \) has no solutions. False, if the determinant of an \( n \times n \) matrix \( A \) is nonzero then the matrix is invertible, which means that \( A x=0 \) has infinite solutions. True, if the determinant of an \( n \times n \) matrix \( A \) is nonzero then the matrix is not invertible, which mean True, if the determinant of an \( n \times n \) matrix \( A \) is nonzero then the matrix is invertible, which means that \( A \mathbf{x}=0 \) has only the trivial solution