# 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}

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

2020-11-08
Given
The given matrix is
$$B=\begin{bmatrix}2 & 3 \\-4 & 2 \end{bmatrix} , A_1=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} , A_2=\begin{bmatrix}0 &-1 \\1 & 0 \end{bmatrix} , A_3=\begin{bmatrix}1 &1 \\0 & 1 \end{bmatrix}$$
Calculation:
Let us represent B as a linear combination of $$A_1, A_2,A_3$$
$$B=c_1A_1+c_2A_2+c_3A_3$$
$$\begin{bmatrix}2 & 3 \\-4 & 2 \end{bmatrix}=c_1\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}+c_2\begin{bmatrix}0 &-1 \\1 & 0 \end{bmatrix}+c_3\begin{bmatrix}1 &1 \\0 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}c_1 &0 \\0 & c_1 \end{bmatrix}+\begin{bmatrix}0 & -c_2 \\ c_2 & 0 \end{bmatrix}+\begin{bmatrix}c_3 & c_3 \\0 & c_3 \end{bmatrix}$$
$$=\begin{bmatrix}c_1+0+c_3 & 0-c_2+c_3 \\0+c_2+0 & c_1+0+c_3 \end{bmatrix}$$
Step 3
Now , by comparing both sides.
$$c_1+0+c_3=2 \dots(1)$$
$$0-c_2+c_3=3 \dots(2)$$
$$0+c_2+0=-4 \dots(3)$$
$$c_1+0+c_3=2 \dots(4)$$
from equation (3) and (2)
$$c_2=-4 \text{ and } c_3=-1$$
substitute $$c_3=-1 \text{ in (1) we get } c_1=3$$
Thus ,we can say $$B=3A_1-4A_2-A_3$$

### Relevant Questions

write B as a linear combination of the other matrices, if possible.
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A^2B
let $$A=\begin{bmatrix}1 & 2 \\3 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix}2 & 0 & -1 \\3 & -3 & 4 \end{bmatrix}$$
Let $$A=\begin{bmatrix}3 & 0 \\ -1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2 & 1 \\ 0 & 2 &3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 &6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\ -2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} , F=\begin{bmatrix}-1 \\ 2 \end{bmatrix}$$
$$A=\begin{bmatrix}3 & 0 \\-1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2&1 \\0 & 2&3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\3 & 4\\5&6 \end{bmatrix}, D=\begin{bmatrix}0 & -3 \\-2 & 1 \end{bmatrix},E=\begin{bmatrix}4 & 2 \end{bmatrix},F=\begin{bmatrix}-1 \\2 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 0&1 \\ 0 & -1 &0 \\ 0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$