Exercise:
Consider a block matrix M of the type M leftarray@c|c@ matrix A matrix & B hline matrix matrix & C arrayright where Ain M_rtimes rK Cin M_stimes sK Bin M_rtimes sK. Proof that textdetMtextdetA textdetC.

Solution:
Proof. Define DABC:textdet leftarray@c|c@ matrix A matrix & B hline matrix matrix & C arrayright as a function D:M_rtimes rKtimes M_rtimes sKtimes M_stimes sK longrightarrow K. If we fix the first two matrices A and B then the function M_stimes sK longrightarrow K Clongmapsto DABC is s-linear and alternating. By a previous theorem we have DABCtextdetC DABI_s. Let's calculate DABI_s. By definition DABI_s leftarray@c|c@ matrix A matrix & B hline matrix matrix & I_s arrayright By doing row operations of the type R_i+cR_jlongrightarrow R_k jneq i cin K we can use the last s-rows of leftarray@c|c@ matrix A matrix & B hline matrix matrix & I_s arrayright to get to the matrix leftarray@c|c@ matrix A matrix & hline matrix matrix & I_s arrayright. By a previous theorem det leftarray@c|c@ matrix A matrix & B hline matrix matrix & I_s arrayright textdetleftarray@c|c@ matrix A matrix & hline matrix matrix & I_s arrayright. The function M_rtimes rKlongrightarrow K Alongmapsto det leftarray@c|c@ matrix A matrix & hline matrix matrix & I_s arrayright is r-linear and alternating. By a previous result we have det leftarray@c|c@ matrix A matrix & hline matrix matrix & I_s arrayright textdetAtextdet leftarray@c|c@ matrix A matrix & B hline matrix matrix & C arrayright textdetA textdetI_r+s textdetA going back to the results above we get that det leftarray@c|c@ matrix A matrix & B hline matrix matrix & C arrayright DABC:textdetA textdetC.