To determine the value of
|2B(adj(3A))|, we need to utilize some properties of determinants and adjugate matrices.
First, recall that for a square matrix
A of order
n, the adjugate (or adjoint) of
A, denoted as
adj(A) , has a determinant related to the determinant of
A. Specifically, for a
3×3 matrix, this relationship is:
adj(A)=det(A)⋅A−1Thus, the determinant of the adjugate of
A, for a
3×3 matrix, is:
det(adj(A))=det(A)n−1=det(A)2Given
det(A)=, we can find:
det(adj(3A))=(det(3A))2Next, recall the determinant property for a scalar multiple of a matrix:
det(kA)=kn⋅det(A)For
3A, we have:
det(3A)=33⋅det(A)=27⋅= Thus:
det(adj(3A))=()2=Now, for the matrix
2B, use the scalar multiple property again:
det(2B)=23⋅det(B)=8⋅=We now combine these results to find:
|2B(adj(3A))|=det(2B⋅adj(3A))=det(2B)⋅det(adj(3A))Therefore:
|2B(adj(3A))|=⋅=1