$\det(A^2+A-I_2)+\det(A^2+I_2) = 5$

Question

Let $A \in M_{2\times 2}(\mathbb{C})$ and $\det(A)=1\DeclareMathOperator{\tr}{tr}$

Prove that $\det(A^2+A-I_2)+\det(A^2+I_2) = 5$

using Cayley-Hamilton Theorem $A^2-\tr(A)A+\det(A)I_2=0$

$\det\big(\tr(A)A-\det(A)I_2-I_2\big) + \det\big(A(A+A^{-1})\big)=5$

$\det\big(\tr(A)A-I_2(\det(A)+1)\big)+\det(A+A^{-1})=5$

using https://math.stackexchange.com/q/1937052 $\det(A+B)=\det A+\det B+\det A⋅\tr(A^{-1}B) $

$\tr(A)\det(A)-\big(\det(A)+1\big)\det(I_2)+\tr(A)\big(-\det(A)-1\big)^{-1}\tr(I^{-1}A)+\det(A)+\det(A^{-1})+\tr(A^2)=5 $

$\tr(A)-2-\tr(A)^20.5+1+1+\tr(A^2)=5$

$\tr(A)-\tr(A)^20.5+\tr\big(\tr(A)A-\det(A)I_2\big)=5$

$\tr(A)-\tr(A)^20.5+\tr(A)^2-\tr(I_2)=5$

$\tr(A)+0.5\tr(A)^2-2=5$

This is where I am stuck. And also I don't know how to prove the identity which I cited.

Answer 1

You're on the good track.

Let $t=\operatorname{tr}(A)$. Then, from $\det(A)=1$, we know by Cayley-Hamilton that $$ A^2-tA+I_2=0 $$ Thus $$ A^2+A-I_2=tA-I_2+A-I_2=(t+1)A-2I_2 $$ and $$ A^2+I_2=tA $$ Thus $\det(A^2+I_2)=t^2\det(A)=t^2$.

The formula you cite tells you that $$ \det(X+Y)=\det(X)+\det(Y)+\det(X)\operatorname{tr}(X^{-1}Y) $$ and we can apply it to $X=-2I_2$, $Y=(t+1)A$. Thus $$ \det(X)=4,\qquad \det(Y)=(t+1)^2,\qquad X^{-1}Y=-\frac{1}{2}(t+1)A, \qquad \operatorname{tr}(X^{-1}Y)=-\frac{1}{2}t(t+1) $$ Thus the expression on the left-hand side becomes $$ 4+(t+1)^2-2t(t+1)+t^2=5 $$

Answer 2

Let $p(x):=\det(x\,I-A)$ be the characteristic polynomial of $A$, where $I$ is a short-hand notation for $I_2$. Then, show that $$\det(A^2+A-I)+\det(A^2+I)=p\left(\phi\right)\,p\left(\bar{\phi}\right)+p(+\text{i})\,p(-\text{i})\,,$$ where $\phi:=\dfrac{-1+\sqrt{5}}{2}$ and $\bar{\phi}:=\dfrac{-1-\sqrt{5}}{2}$. Since $\det(A)=1$, $p(x)=x^2-bx+1$ for some $b\in\mathbb{C}$.

Because $\phi$ and $\bar{\phi}$ are roots of the polynomial $x^2+x-1$, we have $$p(t)=t^2-bt+1=(1-t)-bt+1=2-(1+b)t$$ for $t\in\{\phi,\bar{\phi}\}$. This shows that $$p(\phi)\,p(\bar{\phi})=\big(2-(1+b)\phi\big)\,\big(2-(1+b)\bar{\phi}\big)=4-2(1+b)(\phi+\bar{\phi})+(1+b)^2\phi\bar{\phi}\,.$$ That is, $$p(\phi)\,p(\bar{\phi})=4-2(1+b)(-1)+(1+b)^2(-1)=5-b^2\,.$$

Similarly, $+\text{i}$ and $-\text{i}$ are roots of the polynomial $x^2+1$, we have $$p(t)=t^2-bt+1=(-1)-bt+1=-bt$$ for $t\in\{+\text{i},-\text{i}\}$. Hence, $$p(+\text{i})\,p(-\text{i})=(-b\text{i})(+b\text{i})=b^2\,.$$ The claim follows immediately.

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Below is a proof of the cited identity, i.e., $$\det(A+B)=\det(A)+\det(B)+\det(A)\,\text{tr}(A^{-1}B)$$ for any $A,B\in\text{Mat}_{2\times 2}(K)$ such that $A$ is invertible. Here, $K$ is any field.

Since $A$ is invertible, $$\det(A+B)=\det(A)\,\det(I+A^{-1}B)\,.$$ The characteristic polynomial of $A^{-1}B$ is $$q(x):=\det(x\,I-A^{-1}B)=x^2-\text{tr}(A^{-1}B)\,x+\det(A^{-1}B)\,.$$ Thus, $$\det(I+A^{-1}B)=(-1)^2\,\det(-I-A^{-1}B)=q(-1)=1+\text{tr}(A^{-1}B)+\det(A^{-1}B)\,.$$ Ergo, $$\begin{align}\det(A+B)&=\det(A)\,\Big(1+\text{tr}(A^{-1}B)+\det(A^{-1}B)\Big)\\&=\det(A)+\det(A)\,\det(A^{-1}B)+\det(A)\,\text{tr}(A^{-1}B)\\&=\det(A)+\det(B)+\det(A)\,\text{tr}(A^{-1}B)\,.\end{align}$$

Answer 3

By brute force, calling

$$ A = \left(\begin{array}{cc}a_1 & a_2 \\ a_3 & a_4\end{array}\right) $$

we have

$$ \det A = 1\to a_1 a_4 = 1 + a_2 a_3 $$

and also

$$ \det\left(A^2+A-I_2\right)+\det\left(A^2+I_2\right) = 2 a_4^2 a_1^2+a_4 a_1^2+a_4^2 a_1-a_2 a_3 a_1-4 a_2 a_3 a_4 a_1+a_4 a_1-a_1+2 a_2^2 a_3^2-a_2 a_3-a_2 a_3 a_4-a_4+2 $$

and after substituting $ a_1 a_4 = 1 + a_2 a_3$ we obtain the result

$$ \det\left(A^2+A-I_2\right)+\det\left(A^2+I_2\right) = 5 $$