Question

Compute the elements a₄₃ and a₂₂ of the matrix:
$A=\left[\begin{array}{ccc}0& 1& 0\\ 2& 0& 2\\ 0& 3& 2\\ 4& 0& 4\end{array}\right]\left[\begin{array}{cc}2& -1\\ -3& 2\\ 4& 3\end{array}\right]\left[\begin{array}{ccccc}0& 1& -1& 2& -2\\ 3& -3& 4& -4& 0\end{array}\right]$

Answer 1

We have,

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{ccc}0& 1& 0\\ 2& 0& 2\\ 0& 3& 2\\ 4& 0& 4\end{array}\right]\left[\begin{array}{cc}2& -1\\ -3& 2\\ 4& 3\end{array}\right]\left[\begin{array}{ccccc}0& 1& -1& 2& -2\\ 3& -3& 4& -4& 0\end{array}\right] \\ \Rightarrow A=\left[\begin{array}{ccc}0& 1& 0\\ 2& 0& 2\\ 0& 3& 2\\ 4& 0& 4\end{array}\right]\left[\begin{array}{ccccc}0-3& 2+3& -2-4& 4+4& -4-0\\ 0+6& -3-6& 3+8& -6-8& 6+0\\ 0+9& 4-9& -4+12& 8-12& -8+0\end{array}\right] \\ \Rightarrow A=\left[\begin{array}{ccc}0& 1& 0\\ 2& 0& 2\\ 0& 3& 2\\ 4& 0& 4\end{array}\right]\left[\begin{array}{ccccc}-3& 5& -6& 8& -4\\ 6& -9& 11& -14& 6\\ 9& -5& 8& -4& -8\end{array}\right] \\ \Rightarrow A=\left[\begin{array}{ccccc}0+6+0& 0-9-0& 0+11+0& 0-14-0& 0+6-0\\ -6+0+18& 10-0-10& -12+0+16& 16-0-8& -8+0-16\\ 0+18+18& 0-27-10& 0+33+16& 0-42-8& 0+18-16\\ -12+0+36& 20-0-20& -24+0+32& 32-0-16& -16+0-32\end{array}\right] \\ \Rightarrow A=\left[\begin{array}{ccccc}6& -9& 11& -14& 6\\ 12& 0& 4& 8& -24\\ 36& -37& 49& -50& 2\\ 24& 0& 8& 16& -48\end{array}\right] \\ \therefore a_{43}=8\mathrm{and}{a}_{22}=0 \end{gathered}$$