Vector spaces

Gold Member

Homework Statement

a)A is an invertible 5x5 matrix with complex elements. Find all the values of det(A) where adj(adj(A)) = A. Use the definition of the adjoint (transpose of cofactor matrix)

b) u1,u2,u3 are linearly independent vectors in V. We define:
v1=u1+u2+u3, v2=u1-u2+u3, v3=-u1+3u3-u3
Does Sp{u1,u2,u3}=Sp{v1,v2.v3} ?

Homework Equations

a) The definition of adj(A)
b) If A and B are subsets of space V then Sp(A) = SP(B) iff A is in Sp(B) and B is in Sp(A).

The Attempt at a Solution

a) I tried to use the definition of the adjoint to find the adjoint of adj(A) but it quickly got so complicated that I couldn't see how to calculate det(A) from it. Is there a way to use the fact that A is a 5x5 matrix to simplify things?
b)I think that the answer is yes because obviously {v1,v2,v3} is in the span of {u1,u2,u3} because of their definition. And by solving the equation:
[v1|v2|v3][x]=[un] for all 0<n<=3 I can show that every u is a linear combination of v's and so {u1,u2,u3} is in Sp{v1,v2,v3}. Is that right?
Thanks.

matt grime
Homework Helper
When you say you used the definition of adj(A), do you mearn you used the definition, or the formula? adj(A), is the (unique for det=/=0) matrix satisfying A*adj(A)= det(A)*Id.

Gold Member
I mean the formula: $$[adjA]_{ij} = (-1)^{i+j}*M_{ji}$$
Where $$M_{ji}$$ is the minor (ji). (by minor (j,i) I mean the det of the matrix obtained by taking away the jth row and the ith coloum)

matt grime
Now, what relation does det(A) have to $$[adjA]_{ij} = (-1)^{i+j}*M_{ji}$$?