Let \(A\) and \(B\) be \(n \times n\) matrices over the field \(F\). Show that if \(A\) is invertible there are at most \(n\) scalars \(c\) in \(F\) for which the matrix \(cA + B\) is not invertible.
Solution
Since \(A\) is invertible, \[ cA+B=A(cI+A^{-1}B). \] Hence, by multiplicativity of the determinant, \[ \det(cA+B)=\det(A)\det(cI+A^{-1}B). \] Because \(\det(A)\ne0\), the matrix \(cA+B\) is singular exactly when \[ \det(cI+A^{-1}B)=0. \] The right-hand side is the characteristic polynomial of \(-A^{-1}B\) evaluated at \(c\), and that polynomial has degree \(n\). Hence it has at most \(n\) roots in \(F\). Therefore at most \(n\) scalars \(c\) make \(cA+B\) noninvertible.