We are given that the determinant of a 3rd order matrix A is K. Hence, |A|=K. For a matrix A, the determinant properties used here are: The determinant of the transpose of a matrix is the same as the determinant of the matrix itself So, |AT|=|A|=K. The determinant of the product of a matrix with its transpose is |AAT|=|A||AT|=K×K=K2 Now, we are asked to find the sum of the determinants of matrices A4 and (A−A4). Since the determinant of a product is the product of the determinants and given |A|=K, we have: |A4|=|A|4=K4. To find |A−A4|, note that: By matrix properties, |(A−A4)|= simplify or evaluate separately. Given the expresion |(AAT)+(A−AT)| in the context, we simplify: |AAT|+|(A−A4)|=K2+|A|−|AT|=K2+K−K=K2. Thus the result for the sum of these determinants aligns with the property calculations stated, giving us the simplified determinant result of K2.