GO Classes DPP | GATE CS | Linear Algebra | Eigenvalues of Matrix Products

Answer 1

A standard result says that for any compatible matrices $X$ and $Y$, the matrices $XY$ and $YX$ have the same non-zero eigenvalues, counting algebraic multiplicities.

Now take $X=AB$ and $Y=C$. Then $XY=ABC$ and $YX=CAB$. So $ABC$ and $CAB$ have the same non-zero eigenvalues.

Next, take $X=A$ and $Y=BC$. Then $XY=ABC$ and $YX=BCA$. So $ABC$ and $BCA$ have the same non-zero eigenvalues.

Hence $ABC$, $BCA$, and $CAB$ all have the same non-zero eigenvalues. Therefore, B is necessarily true.

A is false in general. For example, let $A=\begin{bmatrix}1\\0\end{bmatrix}$, $B=\begin{bmatrix}1&0\end{bmatrix}$, and $C=\begin{bmatrix}0&0\\1&0\end{bmatrix}$. Then $ABC=0$, $BCA=0$, but $CAB=\begin{bmatrix}0&0\\1&0\end{bmatrix}$, so the ranks are not all equal.

C is false in general, because these matrices may even have different sizes, so they cannot always be similar.

D is false. For example, let $A=\begin{bmatrix}0&1\\0&0\end{bmatrix}$, $B=\begin{bmatrix}1&0\\0&0\end{bmatrix}$, and $C=\begin{bmatrix}1&0\\0&0\end{bmatrix}$. Then $ABC=0$, but $BCA=\begin{bmatrix}0&1\\0&0\end{bmatrix}\neq0$.

Comments

A) For rank to be same the elements should formed for each ABC,CAB,BCA  matrix should be same but that is not possible here.
C) This is not possible too as they can't be similar at the same time
D) This also points to some sort of similarity which might be true might be not.
😎 Option B is true on general sense as every identical non-zero eigen value can be formed by taking any identical numbers for each matrix given