Discussion
Suppose the two matrices A and B have same characteristic polynomial. Then :
Answer: D
The conclusion that always follows is A is invertible if and only if B is invertible.
The characteristic polynomial \(\det(xI - A)\) has constant term \((-1)^{n}\det A\). Equal characteristic polynomials therefore give \(\det A = \det B\), so one determinant is zero exactly when the other is. Equivalently, 0 is a root of the shared polynomial for both or for neither.
The other options fail. The matrices \(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\) and the zero matrix both have characteristic polynomial \(x^{2}\), but minimal polynomials \(x^{2}\) and \(x\); they are not similar, and neither is the transpose of the other. Same characteristic polynomial says nothing about the Jordan structure, so similarity, equal minimal polynomials and \(A = B^{t}\) are all too strong.
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