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Let $A$ be a matrix, not necessarily square. Say $\mathbf{V}$ and $\mathbf{W}$ are particular solutions of the equations $A \mathbf{V}=\mathbf{Y}_{1}$ and $A \mathbf{W}=\mathbf{Y}_{2}$, respectively, while $\mathbf{Z} \neq 0$ is a solution of the homogeneous equation $A \mathbf{Z}=0$. Answer the following in terms of $\mathbf{V}, \mathbf{W}$, and $\mathbf{Z}$.

a) Find some solution of $A \mathbf{X}=3 \mathbf{Y}_{1}$.
b) Find some solution of $A \mathbf{X}=-5 \mathbf{Y}_{2}$.
c) Find some solution of $A \mathbf{X}=3 \mathbf{Y}_{1}-5 \mathbf{Y}_{2}$.
d) Find another solution (other than $\mathbf{Z}$ and 0 ) of the homogeneous equation $A \mathbf{X}=0$.
e) Find two solutions of $A \mathbf{X}=\mathbf{Y}_{1}$.
f) Find another solution of $A \mathbf{X}=3 \mathbf{Y}_{1}-5 \mathbf{Y}_{2}$.
g) If $A$ is a square matrix, then $\operatorname{det} A=?$
h) If $A$ is a square matrix, for any given vector $\mathbf{W}$ can one always find at least one solution of $A \mathbf{X}=\mathbf{W}$ ? Why?
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