Question: Matrix cancellation. Suppose the scalars a, x, and \(y\) satisfy \(a x=a y\). When \(a \neq 0\) we can conclude that \(x=y\); that is, we can cancel the \(a\) on the left of the equation. In this exercise we explore the matrix analog of cancellation, specifically, what properties of \(A\) are needed to conclude \(X=Y\) from \(A X=A Y\), for matrices A, X, and Y

1. Give an example showing that \(A \neq 0\) is not enough to conclude that \(X=Y\).
2. Show that if \(A\) is left-invertible, we can conclude from \(A X=A Y\) that \(X=Y\).
3. Show that if \(A\) is not left-invertible, there are matrices \(X\) and \(Y\) with \(X \neq Y\), and \(A X=A Y\).

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