Question

Let A = $\left[\begin{array}{ccc}-1& 1& -1\\ 3& -3& 3\\ 5& 5& 5\end{array}\right]$ and B = $\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]$, compute A² − B².

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{ccc}-1& 1& -1\\ 3& -3& 3\\ 5& 5& 5\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{ccc}-1& 1& -1\\ 3& -3& 3\\ 5& 5& 5\end{array}\right]\left[\begin{array}{ccc}-1& 1& -1\\ 3& -3& 3\\ 5& 5& 5\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}1+3-5& -1-3-5& 1+3-5\\ -3-9+15& 3+9+15& -3-9+15\\ -5+15+25& 5-15+25& -5+15+25\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}-1& -9& -1\\ 3& 27& 3\\ 35& 15& 35\end{array}\right] \\ {B}^2=BB \\ \Rightarrow B^2=\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right] \\ \Rightarrow B^2=\left[\begin{array}{ccc}0+4-3& 0-12+12& 0-12+12\\ 0-3+3& 4+9-12& 3+9-12\\ 0+4-4& -4-12+16& -3-12+16\end{array}\right] \\ \Rightarrow B^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ {A}^2-B^2 \\ \Rightarrow A^2-B^2=\left[\begin{array}{ccc}-1& -9& -1\\ 3& 27& 3\\ 35& 15& 35\end{array}\right]-\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \Rightarrow A^2-B^2=\left[\begin{array}{ccc}-1-1& -9-0& -1-0\\ 3-0& 27-1& 3-0\\ 35-0& 15-0& 35-1\end{array}\right] \\ \Rightarrow A^2-B^2=\left[\begin{array}{ccc}-2& -9& -1\\ 3& 26& 3\\ 35& 15& 34\end{array}\right] \end{gathered}$$