Find determinant and trace of product of non square matrices

You need to work with this:

The trace of a matrix is the sum of the eigenvalues and the determinant is the product.

It is not hard to see that the eigenvalues of $BC$ are $2,3,0$. Let $v_1,v_2$ be the eigenvectors correspond to $2,3$ (you will see why we can't use the eigenvector for $0$ soon).

Consider the matrix $CB$ and look at the vectors $Cv_1,Cv_2$. Then $$CB(Cv_i) = C(BCv_i)=\lambda_i Cv_i$$ therefore since $Cv_i\not = 0$ (otherwise $BCv_i=0$) we have that $2,3$ are eigenvalues of $CB$ which means that the determinant is $6$ and the trace is $5$.


I have just come across this old question and I will post my simpler solution to it.
From the Cayley-Hamilton theorem for $CB$ we have that:
$(CB)^2-\operatorname{Tr}(CB)\cdot CB+\det(CB)\cdot I_2=O_2$
Now left multiply by $B$ and then right multiply by $C$ this relation and get that
$(BC)^3-\operatorname{Tr}(CB) \cdot (BC)^2 +\det(CB) \cdot BC =O_3$.
Now simply substitute $BC$'s powers into this equation to get that $\operatorname{Tr}(CB)=5$ and $\det(CB)=6$.

Tags:

Linear Algebra

Matrices

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