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第一个公式:
第二个公式:
证明如下:
[egin{array}{l}
left| {{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}
ight| = left| {egin{array}{*{20}{c}}
{{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}&0\
0&{{I_n}}
end{array}}
ight|\
mathop {{
m{ = = = = = = = = = = }}}limits^{youchengleft| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&0\
{{B_{n imes p}}}&{{I_n}}
end{array}}
ight|} left| {egin{array}{*{20}{c}}
{{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}&0\
{{B_{n imes p}}}&{{I_n}}
end{array}}
ight|\
mathop {{
m{ = = = = = = = = = = = }}}limits^{zuochengleft| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&{ - {C_{p imes n}}A_{n imes n}^{ - 1}}\
0&{{I_n}}
end{array}}
ight|} left| {egin{array}{*{20}{c}}
{{I_{p imes p}} + {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}} - {C_{p imes n}}A_{n imes n}^{ - 1}{B_{n imes p}}}&{ - {C_{p imes n}}A_{n imes n}^{ - 1}}\
{{B_{n imes p}}}&{{I_n}}
end{array}}
ight|{
m{ = }}left| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&{ - {C_{p imes n}}A_{n imes n}^{ - 1}}\
{{B_{n imes p}}}&{{I_n}}
end{array}}
ight|\
mathop {{
m{ = = = = = = = = = = = }}}limits^{youchengleft| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&{{C_{p imes n}}A_{n imes n}^{ - 1}}\
0&{{I_n}}
end{array}}
ight|} left| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&{{C_{p imes n}}A_{n imes n}^{ - 1} - {C_{p imes n}}A_{n imes n}^{ - 1}}\
{{B_{n imes p}}}&{{I_n}{
m{ + }}{B_{n imes p}}{C_{p imes n}}A_{n imes n}^{ - 1}}
end{array}}
ight|{
m{ = }}left| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&0\
{{B_{n imes p}}}&{{I_n}{
m{ + }}{B_{n imes p}}{C_{p imes n}}A_{n imes n}^{ - 1}}
end{array}}
ight|\
{
m{ = }}left| {egin{array}{*{20}{c}}
{{I_{p imes p}}}&0\
{{B_{n imes p}}}&{{I_n}{
m{ + }}{B_{n imes p}}{C_{p imes n}}}
end{array}}
ight| cdot left| {A_{n imes n}^{ - 1}}
ight|{
m{ = }}left| {A_{n imes n}^{ - 1}}
ight| cdot left| {{A_{n imes n}} + {B_{n imes p}}{C_{p imes n}}}
ight|
end{array}]