91Ó°ÊÓ

We are given that \(C = A - B\). So our first step is to establish that this is the matrix we're working with.

We are given that \(A\mathbf{x}_0 = B\mathbf{x}_0\). We can rewrite this expression using matrix C as follows: \(A\mathbf{x}_0 - B\mathbf{x}_0 = \mathbf{0}\) Now, we apply the definition of matrix C: \((A - B)\mathbf{x}_0 = C\mathbf{x}_0 = \mathbf{0}\)

We've reached the expression \(C\mathbf{x}_0 = \mathbf{0}\). Now we need to connect this expression to the property of matrix C being singular. A matrix is singular if its determinant is zero. In other words, the matrix doesn't have an inverse. Another way to express this is stating that the matrix has a non-trivial nullspace, which means there is a nonzero vector \(\mathbf{y}\) such that \(C\mathbf{y} = \mathbf{0}\). From our previous steps, we found that \(C\mathbf{x}_0 = \mathbf{0}\), and we know that \(\mathbf{x}_0 \neq \mathbf{0}\). So, we have found a nonzero vector \(\mathbf{x}_0\) that is in the nullspace of \(C\). Since we found a nonzero vector \(\mathbf{x}_0\) for which \(C\mathbf{x}_0 = \mathbf{0}\), it must be the case that \(C\) is singular.