Question

If $A=\left[\begin{array}{ccc}4& -5& -2\\ 5& -4& 2\\ 2& 2& 8\end{array}\right]$, then adj. (A) equals:

• A. $\left[\begin{array}{ccc}36& -36& 18\\ 36& 36& -18\\ 18& -18& 9\end{array}\right]$
• B. $\left[\begin{array}{ccc}-36& 36& -18\\ -36& 36& -18\\ 18& -18& 9\end{array}\right]$
• C. $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
• D. None of these

Answer 1

Answer: (B)

$A=\left[\begin{array}{ccc}4& -5& -2\\ 5& -4& 2\\ 2& 2& 8\end{array}\right]$
$C_{11}=(-1{)}^2\left|\begin{array}{cc}-4& 2\\ 2& 8\end{array}\right|=-32-4=-36$
$C_{12}=(-1{)}^3\left|\begin{array}{cc}5& 2\\ 2& 8\end{array}\right||=-(40-4)=-36$
$C_{13}=(-1{)}^4\left|\begin{array}{cc}5& -4\\ 2& 2\end{array}\right|=10+8=18$
$C_{21}=(-1{)}^3\left|\begin{array}{cc}-5& -2\\ 2& 8\end{array}\right|=-(-40+4)=36$
$C_{22}=(-1{)}^4\left|\begin{array}{cc}4& -2\\ 2& 8\end{array}\right|=(32+4)=36$
$C_{23}=(-1{)}^5\left|\begin{array}{cc}4& -5\\ 2& 2\end{array}\right|=-(8+10)=-18$
$C_{31}=(-1{)}^4\left|\begin{array}{cc}-5& -2\\ -4& 2\end{array}\right|=-10-8=-18$
$C_{32}=(-1{)}^5\left|\begin{array}{cc}4& -2\\ 5& 2\end{array}\right|=-(8+10)=-18$
$C_{33}=(-1{)}^6\left|\begin{array}{cc}4& -5\\ 5& -4\end{array}\right|=-16+25=9$
$\therefore adj(A)=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$
$={\left[\begin{array}{ccc}-36& -36& 18\\ 36& 36& -18\\ -18& -18& 9\end{array}\right]}^{\prime }=\left[\begin{array}{ccc}-36& 36& -18\\ -36& 36& -18\\ 18& -18& 9\end{array}\right]$