Final answer:
The product of two matrices is always a square matrix, the sum of two matrices is not always a square matrix, the determinant and inverse of two matrices can differ unless they are scalar multiples or inverses of each other.
Step-by-step explanation:
The first statement is true. The product of two $n x n$ matrices, $a$ and $b$, is always a square matrix with dimensions $n x n$. This is because the number of columns in matrix $a$ must be equal to the number of rows in matrix $b$ in order for the product to be defined.
The second statement is not always true. The sum of two $n x n$ matrices, $a$ and $b$, is only a square matrix when the dimensions of $a$ and $b$ are the same (i.e., $n x n$).
The third statement is not always true. The determinant of matrices $a$ and $b$ can be different unless $a$ and $b$ are scalar multiples of each other. In that case, their determinants will be the same.
The fourth statement is not always true. The inverse of matrices $a$ and $b$ can be different unless $a$ and $b$ are inverses of each other. In that case, their inverses will be the same.