Answer the following in terms of \(\mathbf{V}, \mathbf{W}\), and \(\mathbf{Z}\).

Question

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?