Simplifying Determinants Of Matrices: How To Calculate Determinants Of Transpose And Higher Powers Of A Matrix

for any n x n matrix A detA^tdetA

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We can begin by using the properties of the determinant to simplify the given expression:

det(A^T) = det(A)

This is because the transpose of a matrix simply swaps its rows and columns, so the determinant is unchanged.

det(A^T)·det(A) = det(A)·det(A)

Now, we can use another property of the determinant:

det(AB) = det(A)·det(B)

If we set A and B both equal to the original matrix A, we get:

det(AA) = det(A)·det(A)

But AA is just A^2, so we can rewrite this as:

det(A^2) = (det(A))^2

Finally, we can substitute this expression back into the original equation:

det(A^T)·det(A) = det(A^2)

det(A)·det(A) = det(A^2)

Taking the square root of both sides, we get:

det(A) = ±sqrt(det(A^2))

Since A is an n x n matrix, A^2 will be an n x n matrix as well. Therefore, we can iterate this process until we get to A^(2k) for some k, at which point we can simply take the square root k times to get det(A).

Alternatively, we can use other techniques to calculate the determinant, such as finding the LU decomposition or using Gaussian elimination.

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