Question

prove that det(A) = det(B)

Answer 1

if and only if
\[det(A^{T}B^{2}) = det(A^{2}B^{T})\]

Answer 2

use the fact that the det of a product of square matrices is the product of the determinants [ie... \(\det(AB)=\det(A)\det(B)\)] and that \(\det(A^{T})=\det(A)\)

Answer 3

(<=) If det(A^t B²) = det(A² B^t), then det A = det B
Use the properties Zarkon said:
i) det(A^t B²) = det(A^t)*det(B²) = det(A)*det(BB) = det(A)*det(B)*det(B)

ii) det (A² B^t) = det(A²)*det(B^t) = det(AA)*det(B) = det(A)*det(A)*det(B)

Now, equate i) and ii): det(A)*det(B)*det(B) = det(A)*det(A)*det(B)
Cancel out common terms on both sides (Since the determinant is just a number, you could think you have xyy = xxy) and you get det(A) = det(B)

Now, we do the other side,
(=>) If det(A) = det(B), then det(A^t B²) = det(A² B^t)

iii) We can start saying det(A^3) = det(A^3)
iv) det(A^3) = det(A²)*det(A) = det(A²)*det(B)
v) det(A)*det(A²) = det(A²)*det(B)
vi) det(A)*det(B²) = det(A²)*det(B)
vii) det(A^t)*det(B²) = det(A²)*det(B^t)
viii) det(A^t B²) = det(A² B^t)

Q.E.D.

Answer 4

Just recall that in an if or only if statement, you have to prove both sides of the statement.

Answer 5

thank you!!