Question

If the equation$\mathit{d}\mathit{e}\mathit{t}\mathbf{}\mathbf{A}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{}\mathbf{B}$holds for two n x n matrices A and B , is A necessarily similar to B?

Short answer

Therefore, A and B are not similar.

Step-by-step solution

Matrix Definition 

Matrix is a set of numbers arranged in rows andcolumns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a “ m by n” matrix, written “ mxn.”

To find if A is similar to B

$$\begin{gathered} \mathrm{Let}\mathrm{n}=2, \\ \mathrm{A}=\left[\begin{array}{cc}4& 0\\ 0& 1\end{array}\right] \\ \mathrm{B}=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right] \\ \mathrm{Clearly}, \\ \det \mathrm{A}=\det \mathrm{B}=4. \\ \mathrm{Assume},\mathrm{A}\mathrm{and}\mathrm{B}\mathrm{are}\mathrm{similar}. \\ \mathrm{Then}, \\ \exists \mathrm{T}\in {\mathrm{ℝ}}^{2\mathrm{x}2},{\mathrm{T}}^{-1}\mathrm{AT}=\mathrm{B} \\ \mathrm{A}={\mathrm{TBT}}^{-1} \\ \mathrm{A}=2{\mathrm{TIT}}^{-1} \\ \mathrm{A}=2\mathrm{I} \\ \mathrm{A}\ne \mathrm{A}. \\ \mathrm{We}\mathrm{got}\mathrm{a}\mathrm{contradiction}. \\ \mathrm{So},\mathrm{A}\mathrm{and}\mathrm{B}\mathrm{are}\mathrm{not}\mathrm{similar}. \end{gathered}$$