Question

[tex] For A=\left[\begin{array}{ccc}p&q\\r&s\end{array}\right] ,Show that adj(adjA)=A . [\tex]

Answer 1

Answer:

sorry I don't know

Answer 2

Proved $Adj(Adj(A)) = A$

Explanation:

$A = \begin{bmatrix}p & q\\ r & s\end{bmatrix}$

Cofactors of matrix A are

$C_{11} = (-1)^{(1+1)}s = s$

$C_{12} = (-1)^{(1+2)}r =-r$

$C_{21} = (-1)^{(2+1)}q =-q$

$C_{22} = (-1)^{(2+2)}q = p$

$Cof(A)\begin{bmatrix}s &-r \\ -q &p \end{bmatrix}$

$Adj(A) = \left [ Cof(A) \right ]^{t}$

$Adj(A) = \begin{bmatrix}s &-q \\ -r&p\end{bmatrix}$

$Similarly,$

$Adj(Adj(A)) = \begin{bmatrix}p & q\\ r & s\end{bmatrix}$

Hence, $Adj(Adj(A)) = A$