Compute the elements a43 and a22 of the matrix:

Question:

Compute the elements a43 and a22 of the matrix:

$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 2 & 0 & 2 \\ 0 & 3 & 2 \\ 4 & 0 & 4\end{array}\right]\left[\begin{array}{rr}2 & -1 \\ -3 & 2 \\ 4 & 3\end{array}\right]\left[\begin{array}{rrrrr}0 & 1 & -1 & 2 & -2 \\ 3 & -3 & 4 & -4 & 0\end{array}\right]$

Solution:

We have,

Given : $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 2 & 0 & 2 \\ 0 & 3 & 2 \\ 4 & 0 & 4\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ -3 & 2 \\ 4 & 3\end{array}\right]\left[\begin{array}{ccccc}0 & 1 & -1 & 2 & -2 \\ 3 & -3 & 4 & -4 & 0\end{array}\right]$

$\Rightarrow A=\left[\begin{array}{lll}0 & 1 & 0 \\ 2 & 0 & 2 \\ 0 & 3 & 2 \\ 4 & 0 & 4\end{array}\right]\left[\begin{array}{ccccc}0-3 & 2+3 & -2-4 & 4+4 & -4-0 \\ 0+6 & -3-6 & 3+8 & -6-8 & 6+0 \\ 0+9 & 4-9 & -4+12 & 8-12 & -8+0\end{array}\right]$

$\Rightarrow A=\left[\begin{array}{lll}0 & 1 & 0 \\ 2 & 0 & 2 \\ 0 & 3 & 2 \\ 4 & 0 & 4\end{array}\right]\left[\begin{array}{ccccc}-3 & 5 & -6 & 8 & -4 \\ 6 & -9 & 11 & -14 & 6 \\ 9 & -5 & 8 & -4 & -8\end{array}\right]$

$\Rightarrow A=\left[\begin{array}{ccccc}0+6+0 & 0-9-0 & 0+11+0 & 0-14-0 & 0+6-0 \\ -6+0+18 & 10-0-10 & -12+0+16 & 16-0-8 & -8+0-16 \\ 0+18+18 & 0-27-10 & 0+33+16 & 0-42-8 & 0+18-16 \\ -12+0+36 & 20-0-20 & -24+0+32 & 32-0-16 & -16+0-32\end{array}\right]$

$\Rightarrow A=\left[\begin{array}{ccccc}6 & -9 & 11 & -14 & 6 \\ 12 & 0 & 4 & 8 & -24 \\ 36 & -37 & 49 & -50 & 2 \\ 24 & 0 & 8 & 16 & -48\end{array}\right]$

$\therefore a_{43}=8$ and $a_{22}=0$

 

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