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[tex]

\det(\textrm{id} + AB) = \det(\textrm{id} + BA)

[/tex]

is true even when AB and BA are not the same size. In other words, A and B are not neccessarily square matrices.

For example, if

[tex]

A = \big(A_1,\; A_2\big),\quad\quad\quad

B = \left(\begin{array}{c} B_1 \\ B_2 \\ \end{array}\right)

[/tex]

then

[tex]

\det(\textrm{id} + AB) = 1 + A_1B_1 + A_2B_2

[/tex]

and

[tex]

\det(\textrm{id} + BA) = \det\left(\begin{array}{cc}

1 + B_1A_1 & B_1 A_2 \\

B_2 A_1 & 1 + B_2 A_2 \\

\end{array}\right)

[/tex]

[tex]

= (1 + B_1A_1)(1 + B_2A_2) - A_1A_2B_1B_2 = 1 + B_1A_1 + B_2A_2

[/tex]

Anyone knowing how to prove the general case?

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# Det(1+AB) = det(1+BA)

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