how did it become |A|^3 ??
#please explain i have an exam within 12 hours
stray stirrup
pine bearBOT
how did it become |A|^3 ??
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pulsar sorrel
det(lambda I3) = lambda^3 because the dimension is 3 (take the product of the values on the diagonal)
now det(A Aadj) = det( det(A) I) = det(A)^3
stray stirrup
if k constant is multiplied by a matrix then we multiply each element with k . Here k is det A so why are we taking out its cube??
pulsar sorrel
because the matrix is (det A) I_3, that is the 3×3 matrix with only (det A) on its diagonal
for a diagonal matrix, its determinant is the product of the entries on the diagonal, hence you get the determinant of (det A) I_3 to be (det A) × (det A) × (det A) = (det A)^3
in general, if A is a n×n matrix, then det (A Aadj) = det (det A I_n) = (det A)^n
it also makes sense when you consider the interpretation of the 3D determinant as the volume. If you rescale a shape by λ, then you multiply its volume by λ³
stray stirrup
ooh that, its determinant is being taken!! damn i was so confused. Thank you so much for explaining this briefly!!