Matrices and Determinants

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 itselfSo, $|A^T| = |A| = K$.The determinant of the product of a matrix with its transpose is$|AA^T| = |A||A^T| = K \times K = K^2$Now, we are asked to find the sum of the determinants of matrices $A^4$ and $(A-A^4)$.Since the determinant of a product is the product of the determinants and given $|A|=K$, we have:$|A^4| = |A|^4 = K^4 .$To find $|A-A^4|$, note that:By matrix properties, $| (A-A^4) | =$ simplify or evaluate separately.Given the expresion $| (AA^T) + (A-A^T) |$ in the context, we simplify:$|AA^T| + |A-A^4| = K^2 + |A| - |A^T| = K^2 + K - K = K^2 .$Thus the result for the sum of these determinants aligns with the property calculations stated, giving us the simplified determinant result of $K^2$.