Prove $\det(I + B) = 2(1 + tr(B)).$

Question

Let A be a $3\times 3$ invertible matrix (with real coefficients) and let $B=A^TA^{-1}$. Prove that

\begin{equation*} \det(I + B) = 2(1 + tr(B)). \end{equation*}

I know that

\begin{equation*} \det (I+B)=\lambda_1\lambda_2\lambda_3+\lambda_1\lambda_2 +\lambda_1\lambda_3+\lambda_2\lambda_3+\lambda_1+\lambda_2+\lambda_3+1 \end{equation*}

given $\lambda_1,\lambda_2$ and $\lambda_3$ are three distinct eigenvalues of $B$. However, I don't know where to go on from here and how to utilise the fact that $B=A^TA^{-1}$. Any help or direction would be appreciated.

Answer 1

Note that $\det B = 1$.

Note additionally that $B^{-1} = A(A^T)^{-1}$, so that $$ \operatorname{trace}(B^{-1}) = \operatorname{trace}(A(A^T)^{-1}) = \operatorname{trace}((A^T)^{-1}A) = \operatorname{trace}(B^T) = \operatorname{trace}(B) $$ Now, your polynomial can be written as $$ \det(I + B) = \det(B) + \det(B)\operatorname{trace}(B^{-1}) + \operatorname{trace}(B) + 1 $$ the conclusion follows.

Answer 2

It suffices to show that $1+\lambda_1+\lambda_2+\lambda_3=\lambda_1\lambda_2\lambda_3+\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1$. But this is true because $\det(A^\top {A^{-1}})=1$, meaning $\lambda_1\lambda_2\lambda_3=1$.

Answer 3

Hint: $tr(A^{-1}A^T)=tr(A^TA^{-1})=tr(A^{-T}A)=tr(B^{-1})$

$\sum \lambda_i = tr(B)$

$\sum \lambda_1\lambda_2 = det(B)tr(B^{-1})$

Answer 4

An alternative approach, which doesn't use the explicit formula in terms of eigenvalues. Instead, find the characteristic polynomial. See that, for any $\lambda$,

$$p(\lambda) = \det (B - \lambda I) = -\lambda^3 + \lambda^2 \mathrm{Tr} \, B - \lambda x + \det B$$

for some number $x$. Here we merely have $\lambda = -1$.

The number $x$ can be computed in a variety of ways. Using exterior algebra is one method: construct the $3 \times 3$ matrix $B_2$, which acts on $3 \times 1$ column vectors corresponding to elements of $\bigwedge^2 \mathbb R^3$. Then $x = \mathrm{Tr} \, B_2$.

The relationship between $B_2$ and $B$ is explicitly

$$B_2 (a \wedge b) = B(a) \wedge B(b)$$

for any vectors $a, b$.

Now, use a common inversion identity:

$$B^{-1}(a) = \star B_2^T(\star a)/\det B$$

where $\star$ is the usual Hodge dual. This means we can write $B_2$ as

$$B_2(a \wedge b) = \star (B^T)^{-1}(\star [a \wedge b]) \det B$$

You can verify now (e.g. by breaking into a basis) that $\mathrm{Tr} \, B_2 = \det B \, \mathrm{Tr} \, (B^T)^{-1}$. By the arguments given in other answers, this is merely $\det B \, \mathrm{Tr} \, B$, and as a result, we have

$$p(\lambda) = \det(B - \lambda I) = -\lambda^3 + \lambda^2 \mathrm{Tr} \, B - \lambda [\det B \, \mathrm{Tr} \, B] + \det B$$

For $\lambda =-1$, and since $\det B = 1$, the result follows.