Question Number 202762 by Mastermind last updated on 02/Jan/24
Answered by shunmisaki007 last updated on 03/Jan/24
![A= [(3,0,0),(0,3,0),(0,0,3) ] adj(A)= [( determinant ((3,0),(0,3)), determinant ((0,0),(0,3)), determinant ((0,0),(3,0))),( determinant ((0,0),(0,3)), determinant ((3,0),(0,3)), determinant ((3,0),(0,0))),( determinant ((0,3),(0,0)), determinant ((3,0),(0,0)), determinant ((3,0),(0,3))) ]= [(9,0,0),(0,9,0),(0,0,9) ] adj(adj(A))= [( determinant ((9,0),(0,9)), determinant ((0,0),(0,9)), determinant ((0,0),(9,0))),( determinant ((0,0),(0,9)), determinant ((9,0),(0,9)), determinant ((9,0),(0,0))),( determinant ((0,9),(0,0)), determinant ((9,0),(0,0)), determinant ((9,0),(0,9))) ]= [((27),0,0),(0,(27),0),(0,0,(27)) ] det(adj(adj(A)))= determinant (((27),0,0),(0,(27),0),(0,( 0),(27)))=19,683 ((det(adj(adj(A))))/5)=((19,683)/5)=3,936(3/5) ∴ {((det(adj(adj(A))))/5)}=(3/5) ★](https://www.tinkutara.com/question/Q202808.png)
$$\boldsymbol{{A}}=\begin{bmatrix}{\mathrm{3}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{3}}\end{bmatrix} \\ $$$$\mathrm{adj}\left(\boldsymbol{{A}}\right)=\begin{bmatrix}{\begin{vmatrix}{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{3}}&{\mathrm{0}}\end{vmatrix}}\\{\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}\end{vmatrix}}\\{\begin{vmatrix}{\mathrm{0}}&{\mathrm{3}}\\{\mathrm{0}}&{\mathrm{0}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}\end{vmatrix}}\end{bmatrix}=\begin{bmatrix}{\mathrm{9}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{9}}\end{bmatrix} \\ $$$$\mathrm{adj}\left(\mathrm{adj}\left(\boldsymbol{{A}}\right)\right)=\begin{bmatrix}{\begin{vmatrix}{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{9}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{9}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{9}}&{\mathrm{0}}\end{vmatrix}}\\{\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{9}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{9}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}\end{vmatrix}}\\{\begin{vmatrix}{\mathrm{0}}&{\mathrm{9}}\\{\mathrm{0}}&{\mathrm{0}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}\end{vmatrix}}&{\begin{vmatrix}{\mathrm{9}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{9}}\end{vmatrix}}\end{bmatrix}=\begin{bmatrix}{\mathrm{27}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{27}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{27}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{adj}\left(\mathrm{adj}\left(\boldsymbol{{A}}\right)\right)\right)=\begin{vmatrix}{\mathrm{27}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{27}}&{\mathrm{0}}\\{\mathrm{0}}&{\:\mathrm{0}}&{\mathrm{27}}\end{vmatrix}=\mathrm{19},\mathrm{683} \\ $$$$\frac{\mathrm{det}\left(\mathrm{adj}\left(\mathrm{adj}\left(\boldsymbol{{A}}\right)\right)\right)}{\mathrm{5}}=\frac{\mathrm{19},\mathrm{683}}{\mathrm{5}}=\mathrm{3},\mathrm{936}\frac{\mathrm{3}}{\mathrm{5}} \\ $$$$\therefore\:\left\{\frac{\mathrm{det}\left(\mathrm{adj}\left(\mathrm{adj}\left(\boldsymbol{{A}}\right)\right)\right)}{\mathrm{5}}\right\}=\frac{\mathrm{3}}{\mathrm{5}}\:\bigstar \\ $$