I have a linear transformation, T, such that;
T:${M_{22}}$→${M_{22}}$:
T$\left(\begin{bmatrix}{x_{11}} & {x_{12}}\\{x_{21}} & {x_{22}}\end{bmatrix} \right)= \begin{bmatrix}{{x_{12}}-5{x_{21}}-{x_{22}}} & {-{x_{11}}-2{x_{12}}+3{x_{21}}+4{x_{22}}}\\{-3{x_{21}}} & {-{x_{11}}-{x_{12}}+{x_{21}}+3{x_{22}}}\end{bmatrix}$
What is the matrix that represents this ${M_{22}}$→${M_{22}}$ transformation?
Is it:
$ \begin{bmatrix}{0} & {1} & {-5} & {-1}\\{-1} & {-2} & {3} & {4}\\{0} & {0} & {-3} & {0}\\{-1} & {-1} & {1} & {3}\end{bmatrix}$
If so, how could this be multiplied by a 2×2 matrix to give another 2×2 matrix. (2×2 matrices cannot multiply with 4×4 matrices).
There is likely something small I am missing so some help would be great!
Thanks
Best Answer
You'd have to write the matrix in column form $\begin{pmatrix} x_{11} \\ x_{12} \\ x_{21} \\ x_{22} \end{pmatrix}$. After that, the matrix you gave will work. No 2x2 matrix will multiply to produce that transformation since the top-left entry contains $x_{22}$, which is impossible under matrix multiplication.