Iterated Adjoints of a Matrix

Narahari Santosh Kumar

03/05/2020 0 0

Iteratively taking the adjugate of an invertible matrix A k times yields

$\overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} )=\det(\mathbf {A} )^{\frac {(n-1)^{k}-(-1)^{k}}{n}}\mathbf {A} ^{(-1)^{k}}~,$

$\det(\overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} ))=\det(\mathbf {A} )^{(n-1)^{k}}~.$

For example,

$\operatorname {adj} (\operatorname {adj} (\mathbf {A} ))=\det(\mathbf {A} )^{n-2}\mathbf {A} .$

$\det(\operatorname {adj} (\operatorname {adj} (\mathbf {A} )))=\det(\mathbf {A} )^{(n-1)^{2}}.$

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