Let A = 1 1 1 and B = 9 2 - 10 2 11 2 12 2 13 2 - 14 2 - 15 2 16 2 17 2 , then the value of A ' B A is;

Given $A=\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}{9}^2& -{10}^2& {11}^2\\ {12}^2& {13}^2& -{14}^2\\ -{15}^2& {16}^2& {17}^2\end{array}\right]$ 

then $A^{\text{'}}=\left[\begin{array}{ccc}1& 1& 1\end{array}\right]$

So $A^{\text{'}}B=\left[\begin{array}{lll}1& 1& 1\end{array}\right]\left[\begin{array}{lcc}{9}^2& -{10}^2& {11}^2\\ {12}^2& {13}^2& -{14}^2\\ -{15}^2& {16}^2& {17}^2\end{array}\right]$

$=\left[\begin{array}{ccc}{9}^2+{12}^2-{15}^2& -{10}^2+{13}^2+{16}^2& {11}^2-{14}^2+{17}^2\end{array}\right]$

$=\left[\begin{array}{ccc}0& 325& 214\end{array}\right]$

Now $A^{\text{'}}BA=\left[\begin{array}{ccc}0& 325& 214\end{array}\right]\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]$

$=\left[325+214\right]$

$=\left[539\right]$