Question

Find 2×2 matrix A that maps (1,3)^T and (1,4)^T into (-2,5)^T and (3,-1)^T, respectively

Answer 1

Solution;

A linear transformation $T_A:R^2\to R^2$ Is defined by;

$T_A(v)=A(v)$

Since A is the desired matrix of transformation,we have;

$A\begin{bmatrix} 1\\ 3 \end{bmatrix}=\begin{bmatrix} -2 \\ 5 \end{bmatrix}$ and $A\begin{bmatrix} 1 \\ 4 \end{bmatrix}=\begin{bmatrix} 3 \\ -1 \end{bmatrix}$

If we put both maps into one matrix ,we have;

$A\begin{bmatrix} 1 & 1\\ 3 & 4 \end{bmatrix}=\begin{bmatrix} -2 & 3 \\ 5 & -1 \end{bmatrix}$

Hence we can find A as;

$A=\begin{bmatrix} -2 & 3 \\ 5 & -1 \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix}^{-1}$

$A=\begin{bmatrix} -2 & 3\\ 5& -1 \end{bmatrix}\begin{bmatrix} 4 & -1 \\ -3 & 1 \end{bmatrix}$

$A=\begin{bmatrix} -17& 5\\ 23 & -6 \end{bmatrix}$