Final answer:
To find the determinant of the transpose of Mat A^2, calculate Mat A^2, then find the transpose of the result and calculate its determinant. The determinant is 10,077,696.
Step-by-step explanation:
To find the determinant of the transpose of Mat A^2, let's first calculate Mat A^2. Mat A squared is obtained by multiplying Mat A by itself: [[1,2,3],[1,2,3],[1,2,3]] * [[1,2,3],[1,2,3],[1,2,3]] = [[6,12,18],[6,12,18],[6,12,18]].
The transpose of Mat A^2 is obtained by interchanging its rows and columns: [[6,6,6],[12,12,12],[18,18,18]].
To calculate the determinant of this matrix, we can use the fact that the determinant of a matrix is equal to the product of its eigenvalues. Since all the rows in this matrix are equal, its eigenvalues will all be equal as well. Therefore, we can simply take the determinant of one of the rows and raise it to the power of the number of rows: (6 * 6 * 6)^(3) = 216^3 = 10,077,696.