Question

LET A and B be any two matrices such that B in the inverse of A.

1) Determine the relationship between the adjoint of A and the adjoint of B (5 MARKS)

2) Determine the relationship between the transpose of A and the transpose of B (5 marks).

Please assist.

Answer 1

$B=A^{-1}$

Let $A^*$ be adjoint of $A$,

$B^*$ be adjoint of $B$.

1)

$A^{-1}=\frac{A^*}{detA}$ and $B^{-1}=\frac{B^*}{detB}$

$B=A^{-1}=\frac{A^*}{detA}$ (multyply by $B^{-1}$ )

$BB^{-1}=\frac{A^*}{detA}B^{-1}$

$E=\frac{A^*}{detA}\frac{B*}{detB}$

$E=\frac{A^*}{detA}\frac{B*}{det(A^{-1})}$

$E=A^*B^*$

$B^*=(A^*)^{-1}$

2)

$A=B^{-1}$

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

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

Another proof:

$AB=E$

$(AB)^T=E^T$

$B^TA^T=E$

$(B^T)^{-1}B^TA^T=(B^T)^{-1}E$

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