Answer to Question #111085 in Linear Algebra for Hetisani Sewela

Question

Let A and B be any two matrices such that B is the inverse of A.
3.1 Determine the relationship between the adjoint of A and the adjoint of B. (5)
3.2 Determine the relationship between the transpose of A and the transpose of B.

Answer 1

Given that , $A^{-1}=B$ .

1) We known that,

$\text{adj}(PQ)=(\text{adj}Q)(\text{adj}P) , \text{for any square matrix } P,Q \ \text{of the same order}.$

And $\text{adj}(I)=I$ ,where $I$ is the identity matrix.

Since $AB=I=BA$

Therefore, $\text{adj}(AB)=adj(I)=adj(BA).$

$\implies (adjB)(adjA)=I=(adjA)(adjB)$

$\implies (adjB)=(adjA)^{-1}$ .

2) We known that $(PQ)^T=Q^TP^T$ for any square matrix $P,Q$ of same order .

Since ,$AB=I=BA$

$\implies (AB)^T=I^T=(BA)^T$

$\implies B^TA^T=I=A^TB^T,$ as $I^T=I$

$\implies B^T={(A^T)}.^{-1}$