To check whether a matrix is linear combination of others, we introduce coeffiencts as a(X1)+b(X2)+c(X3)=X4 but how do we deal if matrices are 2x2? like: To check whether a matrix is linear combination of others, we introduce coeffiencts as $aX_1+bX_2+cX_3=X_4 $ but how do we deal if matrices are 2x2? like:
$$ a\begin{pmatrix} 5 & 7 \\ 5 & 1 \end{pmatrix} + b \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} + c\begin{pmatrix} 3 & 4 \\ 5 & 6 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}. $$
The left hand side becomes
$$\begin{bmatrix} 5a+b+3c&7a+2b+4c\\ 5a+2b+5c&a+3b+6c \end{bmatrix}$$
Equating this to the right hand side, you will get a system of 4 linear equations in three variables.