write B as a linear combination of the other matrices, if possible. B=begin{bmatrix}2 & 3 -4 & 2 end{bmatrix} , A_1=begin{bmatrix}1 & 0 0 & 1 end{bmatrix} , A_2=begin{bmatrix}0 &-1 1 & 0 end{bmatrix} , A_3=begin{bmatrix}1 &1 0 & 1 end{bmatrix}

write B as a linear combination of the other matrices, if possible.
$B=\left[\begin{array}{cc}2& 3\\ -4& 2\end{array}\right],{A}_{1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],{A}_{2}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],{A}_{3}=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
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Given
The given matrix is
$B=\left[\begin{array}{cc}2& 3\\ -4& 2\end{array}\right],{A}_{1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],{A}_{2}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],{A}_{3}=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
Calculation:
Let us represent B as a linear combination of ${A}_{1},{A}_{2},{A}_{3}$
$B={c}_{1}{A}_{1}+{c}_{2}{A}_{2}+{c}_{3}{A}_{3}$
$\left[\begin{array}{cc}2& 3\\ -4& 2\end{array}\right]={c}_{1}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+{c}_{2}\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]+{c}_{3}\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}{c}_{1}& 0\\ 0& {c}_{1}\end{array}\right]+\left[\begin{array}{cc}0& -{c}_{2}\\ {c}_{2}& 0\end{array}\right]+\left[\begin{array}{cc}{c}_{3}& {c}_{3}\\ 0& {c}_{3}\end{array}\right]$
$=\left[\begin{array}{cc}{c}_{1}+0+{c}_{3}& 0-{c}_{2}+{c}_{3}\\ 0+{c}_{2}+0& {c}_{1}+0+{c}_{3}\end{array}\right]$
Step 3
Now , by comparing both sides.
${c}_{1}+0+{c}_{3}=2\dots \left(1\right)$
$0-{c}_{2}+{c}_{3}=3\dots \left(2\right)$
$0+{c}_{2}+0=-4\dots \left(3\right)$
${c}_{1}+0+{c}_{3}=2\dots \left(4\right)$
from equation (3) and (2)

substitute
Thus ,we can say $B=3{A}_{1}-4{A}_{2}-{A}_{3}$
Jeffrey Jordon