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
asked 2020-11-07
write B as a linear combination of the other matrices, if possible.
\(B=\begin{bmatrix}2 & 3 \\-4 & 2 \end{bmatrix} , A_1=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} , A_2=\begin{bmatrix}0 &-1 \\1 & 0 \end{bmatrix} , A_3=\begin{bmatrix}1 &1 \\0 & 1 \end{bmatrix}\)

Answers (1)

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 \(B=3A_1-4A_2-A_3\)
0

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