det⁡(I−AAT)=det⁡(I−ATA)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\det(I-AA^T)=\det(I-A^TA) proven in a dumb way

Let A\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}A be a matrix with entries in a ring, then det⁡(I−AAT)=det⁡(I−ATA)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\det(I-AA^T)=\det(I-A^TA).

When A\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}A is an invertible matrix with entries in a field, AAT\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}AA^T and ATA\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}A^TA are similar. It follows that I−AAT\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}I-AA^T and I−ATA\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}I-A^TA are also similar, and det⁡(I−AAT)=det⁡(I−ATA)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\det(I-AA^T)=\det(I-A^TA).

Specifically, GL⁡n(C)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\GL_n(\mathbb C) contains only solutions to det⁡(I−AAT)=det⁡(I−ATA)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\det(I-AA^T)=\det(I-A^TA), but GL⁡n(C)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\GL_n(\mathbb C) is not some subset of an algebraic curve. Thus det⁡(I−AAT)−det⁡(I−ATA)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\det(I-AA^T)-\det(I-A^TA) is the zero polynomial in A\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}A's entries. Its coefficients can be computed in Z\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\mathbb Z, so det⁡(I−AAT)=det⁡(I−ATA)\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\det(I-AA^T)=\det(I-A^TA) holds for any square matrix.

But for any non-square matrix, we can fill in zeroes to make the shorter side equal in length to the longer side on the right or bottom without changing the value of both sides of the equation, so it actually holds for any matrix.

I suspect there's a better way of proving this, but I never learned it. (insert cirno)