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 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.